Instabilities and chaos in the classical three-body and three-rotor problems
IInstabilities and chaos in the classicalthree-body and three-rotor problems by Himalaya Senapati
A thesis submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Physics toChennai Mathematical InstituteSubmitted: April 2020Defended: July 23, 2020Plot H1, SIPCOT IT Park, Siruseri,Kelambakkam, Tamil Nadu 603103,India a r X i v : . [ n li n . C D ] A ug dvisor:Prof. Govind S. Krishnaswami, Chennai Mathematical Institute.
Doctoral Committee Members:1. Prof. K. G. Arun,
Chennai Mathematical Institute.
2. Prof. Arul Lakshminarayan,
Indian Institute of Technology Madras. eclaration
This thesis is a presentation of my original research work, carried out under the guidance ofProf. Govind S. Krishnaswami at the Chennai Mathematical Institute. This work has notformed the basis for the award of any degree, diploma, associateship, fellowship or other titlesin Chennai Mathematical Institute or any other university or institution of higher education.Himalaya SenapatiApril, 2020In my capacity as the supervisor of the candidate’s thesis, I certify that the above statementsare true to the best of my knowledge. Prof. Govind S. KrishnaswamiApril, 2020iii cknowledgments
I would like to start by thanking my supervisor, Professor Govind S. Krishnaswami for hissupport and guidance, it has been a pleasure to be his student and learn from him. I amgrateful to him for suggesting interesting research directions while giving me necessary spaceto pursue my own reasoning. His office was always open and he was willing to carefully listento arguments, even when it was a different line of thought than his own, and offer correctionsand insights. His persistent effort to improve the presentation of our results so that they areself-contained and appeal to a wide audience is something to strive for. Aside from research,he has also been very kind in other aspects of life, ranging from helping preparing applicationsfor workshops and conferences to taking care of administrative matters.I thank Prof. K. G. Arun who is on my doctoral committee for his encouragement and fororganizing departmental seminars with Prof. Alok Laddha where Research Scholars couldpresent their work. I thank Prof. Arul Lakshminarayan for his insightful comments andreferences to the literature during our doctoral committee meetings. I also extend my thanksto Prof. Athanase Papadopoulos for encouraging me to contribute articles on non-Euclideangeometries to a book and for inviting me to conferences at MFO, Oberwolfach and BHU,Varanasi. I thank Prof. Sudhir Jain for valuable discussions. I would also like to thankProfessors S G Rajeev and MS Santhanam for carefully reading this thesis and for theirquestions and suggestions.I am indebted to Prof. Swadheenananda Pattanayak and Prof. Chandra Kishore Moha-patra for instilling in me a love for mathematical sciences from a young age via Olympiadtraining camps at the state level. I am grateful to Banamali Mishra, Gokulananda Das andRamachandra Hota for their teaching. I also thank Sandip Dasverma for his support.I thank Kedar Kolekar, Sonakshi Sachdev and T R Vishnu for many coffee time discus-sions. I also thank my officemates Abhishek Bharadwaj, Dharm Veer and Sarjick Bakshi aswell as A Manu, Anbu Arjunan, Aneesh P B, Anudhyan Boral, Athira P V, Gaurav Patil,Keerthan Ravi, Krishnendu N V, Miheer Dewaskar, Naveen K, Pratik Roy, Praveen Roy,Priyanka Rao, Rajit Datta, Ramadas N, Sachin Phatak, Shanmugapriya P, Shraddha Sri-vastava and Swati Gupta for their help and friendship. A special thanks to Apolline Louvetfor hosting me in France for a part of my stay during both my visits to Europe.I thank the faculty at the Chennai Mathematical Institute including Professors H S Mani,G Rajasekaran, N D Hari Dass, R Jagannathan, A Laddha, K Narayan, V V Sreedhar, RivParthasarathy, P B Chakraborty, G Date and A Virmani for their time and help. Thanks arealso due to the administrative staff at CMI including S Sripathy, Rajeshwari Nair, G Ranjiniand V Vijayalakshmi as well as the mess, security and housekeeping staff.I gratefully acknowledge support from the Science and Engineering Research Board, Govt.of India in the form of an International Travel Support grant to attend the Berlin Mathe-matical Summer School in 2017, for sponsoring a school on nonlinear dynamics at SPPU,Pune in 2018 and for travel support to attend a CIMPA school at BHU, Varanasi and CNSD2019 at IIT Kanpur. I also acknowledge MFO, Oberwolfach for awarding me an OberwolfachLeibniz Graduate Students travel grant to attend a conference held there. I thank CMI formy research fellowship and for supporting my travel to attend schools and workshops at ICTSBengaluru, MFO Oberwolfach, RKMVERI Belur Math, IISER Tirupati and IIT Kanpur andto meet Professors A Chenciner and L Zdeborova in Paris. I am also grateful to the InfosysFoundation and J N Tata trust for financial support.Finally, I would like to thank my parents Niranjan Senapati and Annapurna Senapatiand brother Meghasan Senapati for their love and support. bstract
This thesis studies instabilities and singularities in a geometrical approach to the planarthree-body problem as well as instabilities, chaos and ergodicity in the three-rotor problem.Trajectories of the three-body problem are expressed as geodesics of the Jacobi-Maupertuis(JM) metric on the configuration space. Translation, rotation and scaling isometries lead toreduced dynamics on quotients of the configuration space, which encode information on thefull dynamics. Riemannian submersions are used to find quotient metrics and to show thatthe geodesic formulation regularizes collisions for the 1 /r , but not for the 1 /r potential.Extending work of Montgomery, we show the negativity of the scalar curvature on the centerof mass configuration space and certain quotients for equal masses and zero energy. Sectionalcurvatures are also found to be largely negative, indicating widespread geodesic instabilities.In the three-rotor problem, three equal masses move on a circle subject to attractivecosine inter-particle potentials. This problem arises as the classical limit of a model ofcoupled Josephson junctions. The energy serves as a control parameter. We find analoguesof the Euler-Lagrange family of periodic solutions: pendula and breathers at all energies andchoreographies up to moderate energies. The model displays order-chaos-order behavior andundergoes a fairly sharp transition to chaos at a critical energy with several manifestations:(a) a dramatic rise in the fraction of Poincar´e surfaces occupied by chaotic sections, (b)spontaneous breaking of discrete symmetries, (c) a geometric cascade of stability transitionsin pendula and (d) a change in sign of the JM curvature. Poincar´e sections indicate globalchaos in a band of energies slightly above this transition where we provide numerical evidencefor ergodicity and mixing with respect to the Liouville measure and study the statistics ofrecurrence times. vi ist of publications This thesis is based on the following publications.1. G. S. Krishnaswami and H. Senapati,
Curvature and geodesic instabilities in a geomet-rical approach to the planar three-body problem , J. Math. Phys., , 102901 (2016).arXiv:1606.05091. [Featured Article]2. G. S. Krishnaswami and H. Senapati, An introduction to the classical three-body prob-lem: From periodic solutions to instabilities and chaos , Resonance, , 87-114 (2019).arXiv:1901.07289.3. G. S. Krishnaswami and H. Senapati, Stability and chaos in the classical three rotorproblem , Indian Academy of Sciences Conference Series, (1), 139 (2019).arXiv:1810.01317.4. G. S. Krishnaswami and H. Senapati, Classical three rotor problem: periodic solutions,stability and chaos , Chaos, (12), 123121 (2019). arXiv:1811.05807. [Editor’s pick,Featured article]5. G. S. Krishnaswami and H. Senapati, Ergodicity, mixing and recurrence in the threerotor problem , Chaos, (4), 043112 (2020). arXiv:1910.04455. [Editor’s pick]vii ontents Acknowledgments ivAbstract viList of publications vii1 Introduction 1 C and quotient spaces R , S and S . . . . 132.2.1.3 Lagrange, Euler, collinear and collision configurations . . . . . . . . 162.2.2 Quotient JM metrics on shape space, S and the shape sphere . . . . . . . . 172.2.2.1 Submersion from C to shape space R . . . . . . . . . . . . . . . . 172.2.2.2 Submersion from R to the shape sphere S . . . . . . . . . . . . . 172.2.2.3 Submersion from C to S and then to S . . . . . . . . . . . . . . 172.2.3 JM metric in the near-collision limit and its completeness . . . . . . . . . . . 182.2.3.1 Geometry near pairwise collisions . . . . . . . . . . . . . . . . . . . 182.2.3.2 Geometry on R and C near triple collisions . . . . . . . . . . . . 202.2.4 Scalar curvature for equal masses and zero energy . . . . . . . . . . . . . . . 222.2.4.1 Scalar curvature on the shape sphere S . . . . . . . . . . . . . . . 232.2.4.2 Scalar curvature on the center-of-mass configuration space C . . . 23 viii ONTENTS ix R and on S . . . . . . . . . . . 242.2.5 Sectional curvature for three equal masses . . . . . . . . . . . . . . . . . . . . 252.2.6 Stability tensor and linear stability of geodesics . . . . . . . . . . . . . . . . . 272.2.6.1 Rotational Lagrange solutions in Newtonian potential . . . . . . . . 282.2.6.2 Lagrange homotheties . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Planar three-body problem with Newtonian potential . . . . . . . . . . . . . . . . . . 292.3.1 JM metric and its curvature on configuration and shape space . . . . . . . . 292.3.2 Near-collision geometry and geodesic incompleteness . . . . . . . . . . . . . . 31 ϕ - ϕ torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Static solutions and their stability . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1.1 Uniformly rotating three-rotor solutions from G, D and T . . . . . . 383.2.1.2 Stability of static solutions . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Changes in topology of Hill region with growing energy . . . . . . . . . . . . 393.2.3 Low and high energy limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.3.1 Three low-energy constants of motion . . . . . . . . . . . . . . . . . 403.3 Reductions to one degree of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Periodic pendulum solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1.1 Stability of pendulum solutions via monodromy matrix . . . . . . . 423.3.1.2 Stability of librational pendula ( E <
4) . . . . . . . . . . . . . . . . 433.3.1.3 Stability of rotational pendula (
E >
4) . . . . . . . . . . . . . . . . 443.3.1.4 Energy dependence of eigenvectors . . . . . . . . . . . . . . . . . . . 453.3.2 Periodic isosceles ‘breather’ solutions . . . . . . . . . . . . . . . . . . . . . . . 463.3.2.1 Linear stability of breathers . . . . . . . . . . . . . . . . . . . . . . . 473.4 Jacobi-Maupertuis metric and curvature . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.1 Behavior of JM curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.2 Geodesic stability of static solutions . . . . . . . . . . . . . . . . . . . . . . . 503.5 Poincar´e sections: periodic orbits and chaos . . . . . . . . . . . . . . . . . . . . . . . 503.5.1 Transition to chaos and global chaos . . . . . . . . . . . . . . . . . . . . . . . 513.5.1.1 Numerical schemes and robustness of Poincar´e sections . . . . . . . 51
CONTENTS ϕ = 0’ Poincar´e surface . . . . . . . . . . . . . . . 543.5.2.1 Pendula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5.2.2 Breathers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5.2.3 A new family of periodic solutions . . . . . . . . . . . . . . . . . . . 553.6 Choreographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6.1 Examples of choreographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.6.2 Non-rotating choreographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.7 Ergodicity in the band of global chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 583.7.1 Distributions along trajectories and over energy hypersurfaces . . . . . . . . . 593.7.2 Approach to ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.8 Mixing in the band of global chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.9 Recurrence time statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.9.1 Exponential law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.9.2 Scale invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.9.3 Loss of memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A.1 Some landmarks in the history of the three-body problem . . . . . . . . . . . . . . . 73A.2 Proof of an upper bound for the scalar curvature . . . . . . . . . . . . . . . . . . . . 78
B Three-rotor problem 81
B.1 Quantum N -rotor problem from XY model . . . . . . . . . . . . . . . . . . . . . . . 81B.2 Positivity of the JM curvature for 0 ≤ E ≤ g . . . . . . . . . . . . . . . . . . . . . . 83B.3 Measuring area of chaotic region on the ‘ ϕ = 0’ Poincar´e surface . . . . . . . . . . . 84B.4 Power-law approach to ergodicity in time . . . . . . . . . . . . . . . . . . . . . . . . 84 hapter 1Introduction The classical three-body problem arose in an attempt to understand the effect of the Sun onthe Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of thebest physicists and mathematicians and led to the discovery of chaos. In the first part ofthis thesis (Chapter 2), we study a geometrical approach to the planar three-body problemsubject to Newtonian or inverse-square potentials and describe results on instabilities andnear collision dynamics by treating trajectories as geodesics of an appropriate metric on theconfiguration space. The second part (Chapter 3) concerns instabilities, chaos and ergodicityin the classical three-rotor problem which we propose as an interesting variant of the three-body problem. It also arises as the classical limit of a model for chains of coupled Josephsonjunctions. Despite the close connections, the two parts are reasonably self-contained andmay be read independently.
The classical gravitational three-body problem [31, 32] is one of the oldest problems in dy-namics and was the place where Poincar´e discovered chaos [17]. It continues to be a fertilearea of research with discovery of new phenomena such as choreographies [14] and Arnolddiffusion [80]. Associated questions of stability have stimulated much work in mechanics andnonlinear and chaotic dynamics [49, 70]. Quantum and fluid mechanical variants with poten-tials other than Newtonian are also of interest, e.g., (a) the dynamics of two-electron atomsand the water molecule [31], (b) the N -vortex problem with logarithmic potentials [65], (c)the problem of three identical bosons with inverse-square potentials (Efimov effect in coldatoms [27, 39]) and (d) the Calogero-Moser system, also with inverse-square potentials [9].The inverse-square potential has some simplifying features over the Newtonian one, due A survey of some landmarks in the history of the three-body problem is presented in Appendix A.1. CHAPTER 1. INTRODUCTION in part to the nature of the scaling symmetry of the Hamiltonian, H ( λ r , λ r , λ r , λ − p , λ − p , λ − p ) = λ − H ( r , r , r , p , p , p ) . (1.1)Here, for a = 1 , r a and p a are position and momentum vectors of the threebodies and λ is a positive real number. As a consequence, the sign of the energy E controlsasymptotic behavior: bodies fly apart or suffer a triple collision according as E is positiveor negative, leaving open the special case E = 0 [68]. Indeed, if m , , are the masses of thethree bodies, the time evolution of the moment of inertia I = (cid:88) a m a r a = (cid:88) a,i m a r ia r ia (1.2)for the inverse square potential is easily obtained from the canonical Poisson brackets { r ai , p bj } = δ ab δ ij : { ˙ I, r ia } = (cid:40)(cid:88) b,j m b r jb ˙ r jb , r ia (cid:41) = (cid:40)(cid:88) b,j r jb p bj , r ia (cid:41) = − r ia and { ˙ I, p aj } = 2 p aj (1.3)implying { ˙ I, T } = 4 T and { ˙ I, V } = 4 V where T = (1 / (cid:80) p a /m a is the kinetic energy and V is the potential energy. Thus, one obtains the Lagrange-Jacobi identity ¨ I = { ˙ I, H } = 4 E where E = T + V is the total conserved energy of the 3 bodies. Consequently, if E > I → ∞ with bodies flying apart while if E < I → I remains non-zero and bounded for all time isparticularly interesting. This happens when initial conditions are chosen so that E = 0 and˙ I = 0. By contrast, for the Newtonian potential, H ( λ − / r , , , λ / p , , ) = λ / H ( r , , , p , , ) (1.4)leads to ¨ I = 4 E − V , which is not sufficient to determine the long-time behavior of I when E < m ij defined by thekinetic energy m ij ( x ) ˙ x i ˙ x j . Indeed, geodesic flow on a compact Riemann surface of constantnegative curvature is a prototypical model for chaos [31]. In the presence of a potential V , trajectories are reparametrized geodesics of the conformally related Jacobi-Maupertuis(JM) metric g ij = ( E − V ( x )) m ij (see [2, 48] and § .1. GEOMETRICAL APPROACH TO THE PLANAR THREE–BODY PROBLEM E >
0, these works suggest that chaos could arise both from negativity of curvature andfrom fluctuations in curvature. Interestingly, the system of three coupled rotors studied inChapter 3 provides a striking connection between a change in sign of the curvature and theonset of widespread chaos.For the planar gravitational three-body problem (i.e. with pairwise Newtonian poten-tials), the JM metric on the full configuration space R ∼ = C has isometries correspondingto translation and rotation invariance groups C and U (1) ( § C ∼ = C / C and shape space R ∼ = C / U(1) [58]. Here, collision configurations are excluded from C and its quotients. When the Newtonian potential is replaced with the inverse-square poten-tial, the zero-energy JM metric acquires a scaling isometry leading to additional quotients: S ∼ = C / scaling and the shape sphere S ∼ = R / scaling (see Fig. 2.2c). Since the collisionconfigurations have been removed, the (non-compact) shape sphere S has the topology of apair of pants and fundamental group given by the free group on two generators. As part ofa series of works on the planar three-body problem, Montgomery [55] shows that for threeequal masses with inverse-square potentials , the curvature of the JM metric on S is negativeexcept at the two Lagrange points, where it vanishes. As a corollary, he shows the uniquenessof the analogue of Moore’s ‘figure 8’ choreography solution (see Fig. A.3b and [59]) up toisometries and establishes that collision solutions are dense within bound ones. In [54, 56], heuses the geometry of the shape sphere to show that zero angular momentum negative energysolutions (other than the Lagrange homotheties ) of the gravitational three-body problemhave at least one syzygy .We begin by extending some of Montgomery’s results on the geometry of the shapesphere to the center-of-mass configuration space C (without any restriction on angularmomentum) and its quotients. In § § § C and its quotients R and S for arbitrary masses and allowed energies. The estimates showing completeness on C are similar to those showing that the classical action (integral of Lagrangian) diverges forcollisional trajectories. In a private communication, Montgomery points out that this wasknown to Poincar´e and has been rediscovered several times (see for example [15, 53, 59]).Completeness establishes that the geodesic reformulation ‘regularizes’ pairwise and triplecollisions by reparametrizing time so that any collision occurs at t = ∞ . In contrast with The 1 /r force corresponding to the inverse-square potential is sometimes called a ‘strong’ force. In a Lagrange homothety, three bodies always occupy vertices of an equilateral triangle which shrinks toa triple collision at the center of mass without rotation. A syzygy is an instantaneous configuration where the three bodies are collinear.
CHAPTER 1. INTRODUCTION other regularizations [11, 81], this does not involve an extrapolation of the dynamics past acollision nor a change in dependent variables. Unlike for the inverse-square potential, we showthat geodesics for the Newtonian potential can reach curvature singularities (binary/triplecollisions) in finite geodesic time ( § less singular than the inverse-square potential and masses collide sooner underNewtonian evolution in the inverse-square potential. However, due to the reparametrizationof time in going from trajectories to geodesics, masses can collide in finite geodesic time inthe Newtonian potential while taking infinitely long to do so in the inverse-square potential.Indeed, for the attractive 1 /r n potential, the JM line-element leads to estimates ∝ (cid:82) η dηη n/ and (cid:82) r drr n/ for the distances to binary and triple collisions from a nearby location ( § n ≥ n < C and its quotients. For the inverse-square potential,we obtain strictly negative upper bounds for scalar curvatures on C , R and S ( § C using the more easily determined ones on its Riemannian quotients; they are found to belargely negative ( § C is strictly negative, though it can have either sign on shape space R ( § → −∞ at collision points.We also discuss the geodesic instability of Lagrange rotation and homothety solutions forequal masses ( § In the classical three-rotor problem, three point particles of equal mass m move on a circlesubject to attractive cosine inter-particle potentials of strength g (see Fig. 1.1a). The prob-lem of two rotors reduces to that of a simple pendulum while the three-rotor system bearssome resemblance to a double pendulum as well as to the planar restricted three-body prob-lem. However, unlike in the gravitational three-body problem, the rotors can pass througheach other so that there are no collisional singularities. In fact, the boundedness of thepotential also ensures the absence of non-collisional singularities leading to global existence As we will soon see, this is physically reasonable since the rotors occupy distinct sites when the three-rotorproblem is viewed as the classical limit of a chain of coupled Josephson junctions. .2. CLASSICAL THREE–ROTOR PROBLEM CM θ θ θ (a) Three coupled rotors θ θ θ Superconducting segments Tunnel Junctions (b) Chain of Josephson junctions
Figure 1.1: (a) Three coupled classical rotors with angular positions θ , , and center of mass CM.(b) An open chain of three coupled Josephson junctions. A closed chain obtained by connecting thefirst and third segments via a junction may be modeled by the quantum three-rotor problem. and uniqueness of solutions. Despite these simplifications, the dynamics of three (or more)rotors is rich and displays novel signatures of the transition from regular to chaotic motionas the coupling (or energy) is varied.The quantum version of the n -rotor problem is also of interest as it is used to model chainsof coupled Josephson junctions [77] (see Fig. 1.1b). Here, the rotor angles are the phasesof the superconducting order parameters associated to the segments between junctions. It iswell-known that this model for chains of coupled Josephson junctions is related to the XYmodel of classical statistical mechanics [77, 78] (see also Appendix B.1 where we obtain thequantum n -rotor problem from the XY model via a partial continuum limit and a Wickrotation). While in the application to the insulator-to-superconductor transition in arrays ofJosephson junctions, one is typically interested in the limit of large n , here we focus on theclassical dynamics of the n = 3 case.The classical n -rotor problem also bears some resemblance to the Frenkel-Kontorova (FK)model [7]. The latter describes a chain of particles subject to nearest neighbor harmonic andonsite cosine potentials. Despite having different potentials and ‘target spaces’ ( R vs S ),the FK and n -rotor problems both admit continuum limits described by the sine-Gordonfield [7, 72]. The n -rotor problem also bears some superficial resemblance to the Kuramotooscillator model [46]: though the interactions are similar, the equations of motion are ofsecond and first order respectively.Though quite different from our model, certain variants of the three-rotor problem havealso been studied, e.g., (a) chaos in the dynamics of three masses moving on a line segmentwith periodic boundary conditions subject to harmonic and 1d-Coulombic inter-particle po-tentials [45], (b) three free but colliding masses moving on a circle and indications of a lackof ergodicity therein [66], (c) coupled rotors with periodic driving and damping, in connec-tion with mode-locking phenomena [20] and (d) an open chain of three coupled rotors withpinning potentials and ends coupled to stochastic heat baths, in connection to ergodicity [23]. CHAPTER 1. INTRODUCTION
In the center of mass frame of the three-rotor problem, we discover three classes of periodicsolutions: choreographies up to moderate relative energies E and pendula and breathers atall E . The system is integrable at E = 0 and ∞ but displays a fairly sharp transition tochaos around E ≈ g , thus providing an instance of an order-chaos-order transition. Wefind several manifestations of this transition: (a) a geometric cascade of stable to unstabletransition energies in pendula as E → g ± ; (b) a transition in the curvature of the Jacobi-Maupertuis metric from being positive to having both signs as E exceeds four, implyingwidespread onset of instabilities; (c) a dramatic rise in the fraction of the area of Poincar´esurfaces occupied by chaotic trajectories and (d) a breakdown of discrete symmetries inPoincar´e sections present at lower energies. Slightly above this transition, we find evidencefor a band of global chaos where we conjecture ergodic behavior. This is in contrast withthe model of three free but colliding masses moving on a circle [66] discussed above wherenumerical investigations indicated a lack of ergodicity.There are several few degrees of freedom models that display global chaos as well as ergod-icity and mixing. Geodesic flow on a constant negative curvature compact Riemann surfaceis a well-known example [74, 75]. Ballistic motion on billiard tables of certain types includingSinai billiards [76] and its generalization to the Lorentz gas [50] provide other canonical ex-amples. Kicked rotors and the corresponding Chirikov standard map [19] are also conjecturedto display global chaos and ergodicity for certain sufficiently large parameter values [30]. Anattractive feature of the three-rotor system is that, in contrast to these canonical examples,it offers the possibility of studying ergodicity in a continuous time autonomous Hamiltoniansystem of particles without boundaries or specular reflections (rotors can ‘pass through’ eachother without colliding). Interestingly, the center of mass dynamics of three rotors may alsobe regarded as geodesic flow on a 2-torus with non-constant curvature (of both signs) of anappropriate Jacobi-Maupertuis metric (see § § ϕ and ϕ . In § .2. CLASSICAL THREE–ROTOR PROBLEM ϕ - ϕ torus, find all static solutions for the relative motion and discusstheir stability (see Fig. 3.1). The system is also shown to be integrable at zero and infinitelyhigh relative energies E (compared to the coupling g ) due to the emergence of additionalconserved quantities. Furthermore, using Morse theory, we discover changes in the topologyof the Hill region of the configuration space at E = 0, 4 g and 4 . g (see Fig. 3.2).In § E → g ± and shows an accumulation ofstable to unstable transition energies at E = 4 g (see Fig. 3.4). In other words, the largestLyapunov exponent switches from positive to zero infinitely often with the widths of the(un)stable windows asymptotically approaching a geometric sequence as the pendulum energyapproaches 4 g . This accumulation bears an interesting resemblance to the Efimov effect [27]as discussed in § § ϕ - ϕ torus as geodesic flow with respectto the Jacobi-Maupertuis metric. We prove in Appendix B.2 that the scalar curvature isstrictly positive on the Hill region for 0 ≤ E ≤ g but acquires both signs above E = 4 g (seeFig. 3.7) indicating widespread geodesic instabilities as E crosses 4 g . In § E = 4 g as manifested in a rapid rise of the fraction of the area of the energetically allowed ‘Hill’region occupied by chaotic sections (see Fig. 3.12a). This is accompanied by a spontaneousbreaking of two discrete symmetries present in Poincar´e sections below this energy (see Figs.3.9 and 3.10). This transition also coincides with the accumulation of stable to unstabletransition energies of the pendulum family of periodic solutions at E = 4 g . Slightly abovethis energy, we find a band of global chaos 5 . g (cid:46) E (cid:46) . g , where the chaotic sectionsfill up the entire Hill region on all Poincar´e surfaces, suggesting ergodic behavior (see Fig.3.12b). Appendix B.3 summarizes the numerical method employed to estimate the fractionof chaos on Poincar´e surfaces.In § § ϕ , ) and momenta ( p , ) over constant energy hypersurfacesweighted by the Liouville measure. While the joint distribution function of ϕ , is uniform CHAPTER 1. INTRODUCTION on the Hill region of the configuration torus at all energies, the distribution of p (and of p ) shows interesting transitions from the Wigner semi-circular distribution when E (cid:28) g to a bimodal distribution for E > . g (see Fig. 3.15). In the band of global chaos, wefind that distributions of ϕ , and p , along generic (chaotic) trajectories are independentof the chosen trajectory and agree with the corresponding distributions over constant energyhypersurfaces, indicating ergodicity. This agreement fails for energies outside this band. In § (cid:104) cos ϕ (cid:105) t and (cid:104) p (cid:105) t along a generic trajectory over the time interval [0 , T ] approach thecorresponding ensemble averages as a power law ∼ T − / (see Fig. 3.17). This is expectedof an ergodic system where correlations decay sufficiently fast in time as shown in AppendixB.4 (see also [24] for a stochastic formulation).In § even in chaotic regions of the phase space at energies just outsidethis band (see Fig. 3.19b).In § τ follows the exponential law (1 / ¯ τ ) exp( − τ / ¯ τ ) with possible deviations atsmall recurrence times (see Fig. 3.23). Though the mean recurrence/relaxation time ¯ τ varieswith the Liouville volume v of the cell, we find that it obeys the scaling law ¯ τ × v / = τ ∗ .This scaling law is similar to the ones discussed in [29, 62] with the scaling exponent 2 / τ ∗ can varywith the location of the cell center, but does not vary significantly with energy in the bandof global chaos. Finally, we demonstrate a loss of memory by showing that the gaps betweensuccessive recurrence times are uncorrelated.We conclude the thesis with a discussion in Chapter 4. hapter 2Instabilities in the planar three-bodyproblem: A geometrical approach This chapter is based on [40] and [41]. Here, we study the planar three-body problem via ageometrical approach. To set the stage, in § Fermat’s principle in optics states that light rays extremize the optical path length (cid:82) n ( r ( τ )) dτ where n ( r ) is the (position dependent) refractive index and τ a parameter along the path .The variational principle of Euler and Maupertuis (1744) is a mechanical analogue of Fermat’sprinciple [2, 48]. It states that the curve that extremizes the abbreviated action (cid:82) q q p · d q holding energy E and the end-points q and q fixed has the same shape as the Newto-nian trajectory. By contrast, Hamilton’s principle of extremal action (1835) states that atrajectory going from q at time t to q at time t is a curve that extremizes the action.It is well-known that the trajectory of a free particle (i.e., subject to no forces) movingon a plane is a straight line. Similarly, trajectories of a free particle moving on the surface ofa sphere are great circles. More generally, for a mechanical system with configuration space M and Lagrangian L = m ij ( q ) ˙ q i ˙ q j , Lagrange’s equations dp i dt = ∂L∂q i are equivalent to thegeodesic equations with respect to the ‘mass’ or ‘kinetic metric’ m ij on M : m ij ¨ q j ( t ) = −
12 ( m ji,k + m ki,j − m jk,i ) ˙ q j ( t ) ˙ q k ( t ) . (2.1) The optical path length (cid:82) n ( r ) dτ is proportional to (cid:82) dτ /λ , which is the geometric length in units ofthe local wavelength λ ( r ) = c/n ( r ) ν . Here, c is the speed of light in vacuum and ν the constant frequency. CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM
Here, m ij,k = ∂m ij /∂q k and p i = ∂L∂ ˙ q i = m ij ˙ q j is the momentum conjugate to coordinate q i . For instance, the kinetic metric ( m rr = m , m θθ = mr , m rθ = m θr = 0) for a freeparticle moving on a plane may be read off from the Lagrangian L = m ( ˙ r + r ˙ θ ) in polarcoordinates, and the geodesic equations shown to reduce to Lagrange’s equations of motion¨ r = r ˙ θ and d ( mr ˙ θ ) /dt = 0.Remarkably, this correspondence between trajectories and geodesics continues to holdeven in the presence of conservative forces derived from a potential V and follows from arefinement of the Euler-Maupertuis principle due to Jacobi. The shapes of trajectories andgeodesics coincide but the Newtonian time along trajectories is not the same as the arc-lengthparameter along geodesics. Precisely, the equations of motion (EOM) m ki ¨ x i ( t ) = − ∂ k V −
12 ( m ik,j + m jk,i − m ij,k ) ˙ x i ( t ) ˙ x j ( t ) (2.2)may be regarded as reparametrized geodesic equations for the Jacobi-Maupertuis (JM) met-ric, ds = g ij dx i dx j = ( E − V ) m ij dx i dx j (2.3)on the classically allowed ‘Hill’ region E − V ≥
0. Notice that √ (cid:82) ds = (cid:82) pdq = (cid:82) ( L + E ) dt so that the length of a geodesic is related to the classical action of the trajectory. The formulafor the inverse JM metric g ij = m ij / ( E − V ) may also be read off from the time-independentHamilton-Jacobi (HJ) equation ( m ij / E − V )) ∂ i W ∂ j W = 1 by analogy with the rescaledkinetic metric m ij / E appearing in the free particle HJ equation ( m ij / E ) ∂ i W ∂ j W = 1 (seep.74 of [68]). The JM metric is conformal to the kinetic metric and depends parametricallyon the conserved energy E = m ij ˙ x i ˙ x j + V . The geodesic equations¨ x l ( λ ) = − g lk ( g ki,j + g kj,i − g ij,k ) ˙ x i ( λ ) ˙ x j ( λ ) (2.4)for the JM metric reduce to (2.2) under the reparametrization ddλ = 1 σ ddt where σ = ( E − V ) √T . (2.5)Here, T = g ij ˙ x i ˙ x j is the conserved ‘kinetic energy’ along geodesics and equals one-halffor arc-length parametrization. To obtain σ , suppose y i ( t ) is a trajectory and z i ( λ ) thecorresponding geodesic. Then at a point x i = z i ( λ ) = y i ( t ), the velocities are related by σ ˙ z i = ˙ y i leading to T = 12 g ij ˙ z i ˙ z j = E − V m ij ˙ z i ˙ z j = E − V σ m ij ˙ y i ˙ y j = (cid:18) E − Vσ (cid:19) . (2.6)This reparametrization of time may be inconsequential in some cases [e.g. Lagrange rotationalsolutions where σ is a constant since V is constant along the trajectory (see § § § .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL § kr / R = 16 Ek/ (2 E − kr ) of the JM metricon configuration space is non-negative everywhere indicating stability. In the planar Keplerproblem with Hamiltonian p / m − k/r , the gaussian curvature of the JM metric ds = m ( E + k/r )( dr + r dθ ) is R = − Ek/ ( m ( k + Er ) ). R is everywhere negative/positivefor E positive/negative and vanishes identically for E = 0 . This reflects the divergence ofnearby hyperbolic orbits and oscillation of nearby elliptical orbits. Negativity of curvaturecould lead to chaos, though not always, as the hyperbolic orbits of the Kepler problem show.As noted, chaos could also arise from curvature fluctuations [10]. We consider the three-body problem with masses moving on a plane regarded as the complexplane C . Its 6 D configuration space (with collision points excluded) is identified with C . Apoint on C represents a triangle on the complex plane with the masses m , , at its vertices x , , ∈ C . It is convenient to work in Jacobi coordinates (Fig. 2.1) J = x − x , J = x − m x + m x m + m and J = m x + m x + m x M , (2.7)in which the kinetic energy KE = (1 / (cid:80) i m i | ˙ x i | remains diagonal: KE = 12 (cid:88) i M i | ˙ J i | where 1 M = 1 m + 1 m , M = 1 m + 1 m + m (2.8)and M = (cid:80) i m i . The KE for motion about the center of mass (CM) is ( M | ˙ J | + M | ˙ J | ).The moment of inertia about the origin I = (cid:80) i =1 m i | x i | too remains diagonal in Jacobicoordinates ( I = (cid:80) i =1 M i | J i | ), while about the CM we have I CM = M | J | + M | J | .With U = − V = (cid:88) i M, g ) and ( N, h ) are two Riemannian manifolds and f : M → N a surjection (an onto map). Then the linearization df ( p ) : T p M → T f ( p ) N is a surjection between tangent spaces. The vertical subspace V ( p ) ⊆ T p M is defined tobe the kernel of df while its orthogonal complement ker( df ) ⊥ with respect to the metric g is the horizontal subspace H ( p ). f is a Riemannian submersion if it preserves lengths ofhorizontal vectors, i.e., if the isomorphism df ( p ) : ker( df ( p )) ⊥ → T f ( p ) N is an isometry ateach point. The Riemannian submersions we are interested in are associated to quotientsof a Riemannian manifold ( M, g ) by the action of a suitable group of isometries G . Thereis a natural surjection f from M to the quotient M/G . Requiring f to be a Riemanniansubmersion defines the quotient metric on M/G : the inner product of a pair of tangentvectors ( u, v ) to M/G is defined as the inner product of any pair of horizontal preimagesunder the map df .The surjection (cid:0) J , ¯ J , J , ¯ J , J , ¯ J (cid:1) (cid:55)→ (cid:0) J , ¯ J , J , ¯ J (cid:1) defines a submersion from config-uration space C to its quotient C by translations. Linearization of this map d J ( J ) : T J C → T J ( J ) C is the Jacobian matrix d J = . (2.11) T J C is the span of ∂ J , ∂ ¯ J , ∂ J , ∂ ¯ J , ∂ J , ∂ ¯ J and a typical tangent vector a ∂ J + a ∂ ¯ J + a ∂ J + a ∂ ¯ J + a ∂ J + a ∂ ¯ J is represented by the column vector (cid:0) a a a a a a (cid:1) t .The vertical subspace V ( J ) of the submersion is defined to be the kernel of d J ( J ) i.e.the span of ∂ J and ∂ ¯ J . The orthogonal complement of V ( J ) in T J C is the horizontalsubspace H ( J ). H ( J ) is spanned by the four orthogonal vectors ∂ J , ∂ ¯ J , ∂ J and ∂ ¯ J . Forthe map J to be a riemannian submersion, lengths of horizontal vectors must be preserved.A typical horizontal vector is of the form a ∂ J + ¯ a ∂ ¯ J + a ∂ J + ¯ a ∂ ¯ J with norm-square( E + U ) (cid:80) M i a i ¯ a i . This defines the quotient metric on C in coordinates J , ¯ J , J and ¯ J : ds = ( E + U )( M | dJ | + M | dJ | ) . (2.12) .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL Position vectors x , , of masses relative to origin and Jacobi vectors J , , . It is convenient to define rescaled coordinates on C , z i = √ M i J i , in terms of which (2.12)becomes ds = ( E + U )( | dz | + | dz | ). The kinetic energy in the CM frame is KE =(1 / | ˙ z | + | ˙ z | ) while I CM = | z | + | z | . C and quotient spaces R , S and S We now specialize to equal masses ( m i = m ) so that M = m/ M = 2 m/ µ i = 1 / C is seen to be conformal to the flat Euclidean metric via the conformalfactor E + U : ds = (cid:32) E + Gm | z | + 2 Gm | z − √ z | + 2 Gm | z + √ z | (cid:33) (cid:0) | dz | + | dz | (cid:1) . (2.13)Rotations U(1) act as a group of isometries of C , taking ( z , z ) (cid:55)→ (cid:0) e iθ z , e iθ z (cid:1) and leavingthe conformal factor invariant. Moreover if E = 0, then scaling z i (cid:55)→ λz i for λ ∈ R + is alsoan isometry. Thus we may quotient the center-of-mass configuration manifold C successivelyby its isometries. We will see that C / U(1) is the shape space R and C /scaling is S .Furthermore the quotient of C by both scaling and rotations leads to the shape sphere S (see Fig. 2.2c, note that collision points are excluded from C , R , S and S ). Points onshape space R represent oriented congruence classes of triangles while those on the shapesphere S represent oriented similarity classes of triangles. Each of these quotient spaces maybe given a JM metric by requiring the projection maps to be Riemannian submersions. Thegeodesic dynamics on C is clarified by studying its projections to these quotient manifolds.We will now describe these Riemannian submersions explicitly in local coordinates. This isgreatly facilitated by choosing coordinates (unlike z , z ) on C in which the Killing vectorfields (KVFs) corresponding to the isometries point along coordinate vector fields. As wewill see, this ensures that the vertical subspaces in the associated Riemannian submersionsare spanned by coordinate vector fields. Thus we introduce the Hopf coordinates ( r, η, ξ , ξ )on C [61] via the transformation z = re i ( ξ + ξ ) sin η and z = re i ( ξ − ξ ) cos η. (2.14)4 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM Here the radial coordinate r = (cid:112) | z | + | z | = √ I CM ≥ ξ determines the relative orientation of z and z while ξ fixes the orientation of the triangle as a whole. More precisely, 2 ξ is the angle from therescaled Jacobi vector z to z while 2 ξ is the sum of the angles subtended by z and z with the horizontal axis in Fig 2.1. Thus we may take 0 ≤ ξ + ξ ≤ π and 0 ≤ ξ − ξ ≤ π or equivalently, − π ≤ ξ ≤ π and | ξ | ≤ ξ ≤ π − | ξ | . Finally, 0 ≤ η ≤ π/ z and z , indeed tan η = | z | / | z | . When r is held fixed, η, ξ and ξ furnish the standard Hopf coordinates parametrizing the three sphere | z | + | z | = r .For fixed r and η , ξ + ξ and ξ − ξ are periodic coordinates on tori. These tori foliatethe above three-sphere as η ranges between 0 and π/ 2. Furthermore, as shown in § η and 2 ξ are polar and azimuthal angles on the two-sphere obtained as the quotient of S byrotations via the Hopf map. (a) m -m collisionEuler pointsLagrange points SyzygyLinesm -m collisionm -m collision (b) (c) Figure 2.2: (a) The shape sphere is topologically a 2-sphere with the three collision points C , , removed, endowed with the quotient JM metric of negative gaussian curvature. Coordinates andphysical locations on the shape sphere are illustrated. 2 η is the polar angle ( 0 ≤ η ≤ π/ ξ isthe azimuthal angle (0 ≤ ξ ≤ π ). The ‘great circle’ composed of the two longitudes ξ = 0 and ξ = π/ C , , and the Euler points E , , . Lagrange points L , lie on the equator η = π/ 4. The shape space R is a cone on theshape sphere. The origin r = 0 of shape space is the triple collision point. (b) The negativelycurved ‘pair of pants’ metric on the shape sphere S . (c) Flowchart of Riemannian submersions. Let us briefly motivate these coordinates and the identification of the above quotientspaces. We begin by noting that the JM metric (2.13) on C is conformal to the flat Euclideanmetric | dz | + | dz | . Recall that the cone on a Riemannian manifold ( M, ds M ) is theCartesian product R + × M with metric dr + r ds M where r > R + . Inparticular, Euclidean C may be viewed as a cone on the round three sphere S . If S is parameterized by Hopf coordinates η, ξ and ξ , then this cone structure allows us touse r, η, ξ and ξ as coordinates on C . Moreover, the Hopf map defines a Riemannian The Hopf map S → S is often expressed in Cartesian coordinates. If | z | + | z | = 1 defines theunit- S ⊂ C and w + w + w = 1 / / R , then w = (cid:0) | z | − | z | (cid:1) / w + iw = z ¯ z . Using Eq. 2.14, we may express the Cartesian coordinates w i in terms of Hopf .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL S to the round two sphere S . Indeed, if we use Hopf coordinates η, ξ , ξ on S , then the Hopf map takes ( η, ξ , ξ ) (cid:55)→ ( η, ξ ) ∈ S . In general, if M → N is a Riemannian submersion, then there is a natural submersion from the cone on M tothe cone on N . In particular, the Hopf map extends to a Riemannian submersion from thecone on the round S to the cone on the round S , i.e. from Euclidean C to Euclidean R taking ( r, η, ξ , ξ ) (cid:55)→ ( r, η, ξ ). As the conformal factor is independent of rotations, thesame map defines a Riemannian submersion from C with the JM metric to shape space R with its quotient JM metric. Finally, for E = 0, scaling (cid:126)r → λ(cid:126)r defines an isometry of thequotient JM metric on shape space R . Quotienting by this isometry we arrive at the shapesphere S with Montgomery’s ‘pair of pants’ metric. Alternatively, we may quotient C firstby the scaling isometry of its JM metric to get S and then by rotations to get S (see Fig.2.2c).With these motivations, we express the equal-mass JM metric on C in Hopf coordinates[generalization to unequal masses is obtained by replacing Gm h below with ˜ h ( η, ξ ) givenin Eq. (2.37)]: ds = (cid:18) E + Gm h ( η, ξ ) r (cid:19) (cid:0) dr + r (cid:0) dη + dξ − η dξ dξ + dξ (cid:1)(cid:1) . (2.15)It is convenient to write h ( η, ξ ) = v + v + v (2.16)where v = r / ( m | x − x | ) is proportional to the pairwise potential between m and m andcyclic permutations thereof. The v i are rotation and scale-invariant, and therefore functionsonly of η and ξ in Hopf coordinates: v , = 2 (cid:0) η ∓ √ η cos 2 ξ (cid:1) and v = 12 sin η . (2.17)Notice that h → ∞ at pairwise collisions. The v i ’s have the common range 1 / ≤ v i < ∞ with v = 1 / m is at the CM of m and m etc. We also have h ≥ v = v = v , corresponding to Lagrange configurations with masses at vertices of anequilateral triangle. To see this, we compute the moment of inertia I CM in two ways. On theone hand I CM = | z | + | z | = r . On the other hand, for equal masses the CM lies at thecentroid of the triangle defined by masses. Thus I CM is (4 m/ × the sum of the squares of coordinates: 2 w = r cos 2 η, w = r sin(2 η ) cos(2 ξ ) and 2 w = r sin(2 η ) sin(2 ξ ) . Let f : ( M, g ) (cid:55)→ ( N, h ) be a Riemannian submersion with local coordinates m i and n j . Let ( r, m i ) and( r, n j ) be local coordinates on the cones C ( M ) and C ( N ). Then ˜ f : ( r, m ) (cid:55)→ ( r, n ) defines a submersionfrom C ( M ) to C ( N ). Consider a horizontal vector a∂ r + b i ∂ m i in T ( r,m ) C ( M ). We will show that d ˜ f preserves its length. Now, if df ( b i ∂ m i ) = c i ∂ n i then d ˜ f ( a∂ r + b i ∂ m i ) = a∂ r + c i ∂ n i . Since ∂ r ⊥ ∂ m i , || a∂ r + b i ∂ m i || = a + r (cid:107) b i ∂ m i (cid:107) = a + r (cid:107) c i ∂ n i (cid:107) as f is a Riemannian submersion. Moreover a + r (cid:107) c i ∂ n i (cid:107) = (cid:107) a∂ r + c i ∂ n i (cid:107) since ∂ r ⊥ ∂ n i . Thus ˜ f is a Riemannian submersion. CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM the medians, which by Apollonius’ theorem is equal to (3 / × the sum of the squares of thesides. Hence I CM = (cid:80) i =1 r / v i . Comparing, we get (cid:80) i =1 /v i = 3 . Since the arithmeticmean is bounded below by the harmonic mean, h/ v + v + v ) / ≥ (cid:0) v − + v − + v − (cid:1) − = 1 . (2.18) The geometry of the JM metric displays interesting behavior at Lagrange and collision con-figurations on C and its quotients. We identify their locations in Hopf coordinates for equal masses. The Jacobi vectors in Hopf coordinates are J = (cid:114) m re i ( ξ + ξ ) sin η and J = (cid:114) m re i ( ξ − ξ ) cos η. (2.19)At a Lagrange configuration, m , , are at vertices of an equilateral triangle. So | J | = √ | J | / η = π/ 4) and J is ⊥ to J (i.e. ξ = ± π/ 4, the sign being fixed by theorientation of the triangle). So Lagrange configurations L , on C occur when η = π/ ξ = ± π/ r and ξ arbitrary. On quotients of C , L , occur at the imagesunder the corresponding projections. Since 2 η and 2 ξ are polar and azimuthal angles onthe shape sphere, L , are at diametrically opposite equatorial locations (see Figs. 2.2a and2.2b). Collinear configurations (syzygies) occur when J and J are (anti)parallel, i.e. when ξ = 0 or π/ 2, with other coordinates arbitrary. On the shape sphere, syzygies occur on the‘great circle’ through the poles corresponding to the longitudes 2 ξ = 0 and π . Collisionsare special collinear configurations. By C i we denote a collision of particles other than the i th one. So C corresponds to J = 0 which lies at the ‘north pole’ ( η = 0) on S . m and m collide when J = J / η = π/ ξ = 0 at C . Similarly, at C , J = − J / η = π/ ξ = π/ 2. The Euler configurations E i for equal massesare collinear configurations where mass m i is at the midpoint of the other two.Finally, we note that the azimuth and co-latitude ( θ and φ ) [55] are often used as co-ordinates on the shape sphere, so that L , are at the poles while C , , and E , , lie onthe equator. This coordinate system makes the symmetry under permutations of massesexplicit, but is not convenient near any of the collisions (e.g. sectional curvatures can bediscontinuous). On the other hand, our coordinates η and ξ , which are related to θ and φ by suitable rotations, sin φ = cos(2 η − π/ 2) sin(2 ξ ) , cos φ sin θ = cos(2 η − π/ 2) cos(2 ξ ) andcos φ cos θ = sin (cid:16) η − π (cid:17) , are convenient near C but not near E or C , . For instance, sectional curvatures can bediscontinuous, as seen in Fig. 2.5. The neighborhoods of the latter configurations may bestudied by re-ordering the masses. .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL S and the shape sphere C to shape space R Rotations z j → e iθ z j act as isometries of the JM metric (2.15) on C . In the Hopf coordinatesof Eq. (2.14), z = re i ( ξ + ξ ) sin η and z = re i ( ξ − ξ ) cos η, (2.20)rotations are generated by translations ξ → ξ + θ and a discrete shift ξ → ξ + π (mod 2 π ).The shift in ξ rotates z i (cid:55)→ − z i , which is not achievable by a translation in ξ due to itsrestricted range, | ξ | ≤ ξ ≤ π − | ξ | and − π ≤ ξ ≤ π . To quotient by this isometry, wedefine a submersion from C → R taking( r, η, ξ , ξ ) (cid:55)→ ( r, η, ξ ) if ξ ≥ r, η, ξ , ξ ) (cid:55)→ ( r, η, ξ + π ) if ξ < . (2.21)The radial, polar and azimuthal coordinates on R are given by r , 2 η and 2 ξ with m - m collisions occurring on the ray η = 0. Under the linearization of this submersion at a point p ∈ C , V ( p ) is spanned by ∂ ξ and H ( p ) by ∂ r , ∂ η and cos 2 η ∂ ξ + ∂ ξ . These horizontalbasis vectors are mapped respectively to ∂ r , ∂ η and ∂ ξ under the linearization of the map.Requiring lengths of horizontal vectors to be preserved we arrive at the following quotientJM metric on R , conformal to the flat metric on R : ds = (cid:18) E + Gm h ( η, ξ ) r (cid:19) (cid:0) dr + r (cid:0) dη + sin η dξ (cid:1)(cid:1) . (2.22)This metric may also be viewed as conformal to a cone on a round 2 -sphere of radius one-half,since 0 ≤ η ≤ π and 0 ≤ ξ ≤ π are the polar and azimuthal angles. R to the shape sphere S The group R + of scalings ( r, η, ξ ) (cid:55)→ ( λr, η, ξ ) acts as an isometry of the zero-energy JMmetric (2.22) on shape space R . The orbits are radial rays emanating from the origin (andthe triple collision point at the origin, which we exclude). The quotient space R / scaling isthe shape sphere S . We define a submersion from shape space to the shape sphere taking( r, η, ξ ) (cid:55)→ ( η, ξ ). Under the linearization of this map at p ∈ R , V ( p ) = span( ∂ r ). Itsorthogonal complement H ( p ) is spanned by ∂ η and ∂ ξ which project to ∂ η and ∂ ξ on S .Requiring the submersion to be Riemannian, we get the quotient ‘pair of pants’ JM metricon the shape sphere which is conformal to the round metric on a 2-sphere of radius one-half: ds = Gm h ( η, ξ ) (cid:0) dη + sin η dξ (cid:1) . (2.23) C to S and then to S For zero energy, it is also possible to quotient the JM metric (2.15) on C , first by its scalingisometries to get S and then by rotations to arrive at the shape sphere. Interestingly, it8 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM follows from the Lagrange-Jacobi identity that when E and ˙ I vanish, r is constant and themotion is confined to a 3 -sphere embedded in C . To quotient by the scaling isometries( r, η, ξ , ξ ) (cid:55)→ ( λr, η, ξ , ξ ) of C , we define the submersion ( r, η, ξ , ξ ) (cid:55)→ ( η, ξ , ξ ) to S ,with ranges of coordinates as on C . The vertical subspace is spanned by ∂ r while ∂ η , ∂ ξ and ∂ ξ span the horizontal subspace. The latter are mapped to ∂ η , ∂ ξ and ∂ ξ on S .The submersion is Riemannian provided we endow S with the following conformally-roundmetric ds = Gm h ( η, ξ ) (cid:0) dη + dξ − η dξ dξ + dξ (cid:1) . (2.24)Rotations generated by ξ → ξ + θ and ξ → ξ + π (mod 2 π ) act as isometries of thismetric on S . We quotient by rotations to get the metric (2.23) on S via the Riemanniansubmersion defined by( η, ξ , ξ ) (cid:55)→ ( η, ξ ) if ξ ≥ η, ξ , ξ ) (cid:55)→ ( η, ξ + π ) if ξ < . (2.25) The equal-mass JM metric components on center-of-mass configuration space C and itsquotients blow up at two- and three-body collisions. However, we study the geometry inthe neighborhood of collision configurations and show that the curvature remains finite inthe limit. Remarkably, it takes infinite geodesic time for collisions to occur which we showby establishing the geodesic completeness of the JM metric on C and its quotients. Bycontrast, collisions can occur in finite time for the Newtonian three-body evolution. The JMgeodesic flow avoids finite time collisions by reparametrizing time along Newtonian trajecto-ries (see Eq. 2.4). Thus the geodesic reformulation of the inverse-square three-body problem‘regularizes’ pairwise and triple collisions. For equal masses (see § η = 0 (with other coor-dinates arbitrary) while the other two binary collisions occur at C and C (see Fig. 2.2a).Triple collisions occur when r = 0. Unlike for the Newtonian potential, sectional curvatureson coordinate 2-planes are finite at pairwise and triple collisions, though some JM metric(2.15) and Riemann tensor components blow up. It is therefore interesting to study thenear-collision geometry of the JM metric.The geometry of the equal-mass JM metric in the neigborhood of a binary collision isthe same irrespective of which pair of bodies collide. Since Hopf coordinates are particularlyconvenient around η = 0, we focus on collisions between the first pair of masses. Montgomery(see Eq. 3.10c of [55]) studied the near-collision geometry on S and showed that it isgeodesically complete. Let us briefly recall the argument. Expanding the equal-mass S metric (2.23) around the collision point η = 0, we get ds ≈ (cid:18) Gm η (cid:19) (cid:0) dη + 4 η dξ (cid:1) = Gm ρ ( dρ + ρ dχ ) (2.26) .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL ρ = 2 η and χ = 2 ξ . ∂ χ is a KVF, so ‘radial’ curves with constant χ are geodesics.Approaching ρ = 0 along a ‘radial’ geodesic shows that the collision point ρ = 0 is at aninfinite distance ( (cid:112) Gm / (cid:82) ρ dρ/ρ ) from any point ( ρ , χ ) in its neighborhood (0 < ρ (cid:28) S with three collision points excluded) is geodesically complete. To clarify the near-collisiongeometry let dλ = − dρ/ √ ρ or λ = − log( ρ/ρ ) / √ 2. This effectively stretches out theneighborhood of the collision point λ = ∞ . The asymptotic metric ds = Gm ( dλ + dχ / ≤ χ ≤ π and λ ≥ (cid:112) Gm / λ the coordinate along the height and χ the azimuthal angle. Thus the JMmetric looks like that of a semi-infinite cylinder near any of the collision points.More generally, for unequal masses, the near-collision metric (2.26) is ds ≈ Gm m M η (cid:0) dη + 4 η dξ (cid:1) [see Eq . (2 . − . S arises as a Riemannian submersion of R , S and C , the infinite distance tobinary collision points on the shape sphere can be used to show that the same holds on eachof the higher dimensional manifolds. To see this, consider the submersion from (say) C to S . Any curve ˜ γ on C maps to a curve γ on S with l (˜ γ ) ≥ l ( γ ) since the lengths ofhorizontal vectors are preserved. If there was a binary collision point at finite distance on C , there would have to be a geodesic of finite length ending at it. However, such a geodesicwould project to a curve on the shape sphere of finite length ending at a collision point,contradicting its completeness.Thus we have shown that the JM metrics (necessarily of zero energy) on S and S withbinary collision points removed, are geodesically complete for arbitrary masses. On the otherhand, to examine completeness on C and R we must allow for triple collisions as well asnon-zero energy. Geodesic completeness in these cases is shown in § R , S and C in somewhat greater detail by Laurentexpanding the JM metric components around η = 0 and keeping only leading terms. Shape space geometry near binary collisions: The equal-mass shape space metricaround η = 0, in the leading order, becomes ds ≈ Gm η r (cid:0) dr + r (cid:0) dη + 4 η dξ (cid:1)(cid:1) = Gm (cid:18) dr ρ r + dρ ρ + dχ (cid:19) , (2.28)where ρ = 2 η and χ = 2 ξ . We define new coordinates λ and κ by dλ = − dρ/ √ ρ , dκ = dr/r so that ρ = ρ e −√ λ . In these coordinates the collision occurs at λ = ∞ . Theasymptotic metric is ds ≈ Gm (cid:18) ρ e √ λ dκ + dλ + 12 dχ (cid:19) (2.29)0 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM where 0 ≤ χ ≤ π (periodic), λ ≥ −∞ < κ < ∞ . This metric has a constant scalarcurvature of − /Gm . The sectional curvature in the ∂ λ − ∂ κ plane is equal to − /Gm , itvanishes in the other two coordinate planes. These values of scalar and sectional curvaturesagree with the limiting values at the 1 -2 collision point calculated for the full metric onshape space. The near-collision topology of shape space is that of the product manifold S χ × R + λ × R κ . Near-collision geometry on C : The equal-mass JM metric in leading order around η = 0is ds ≈ Gm η r (cid:0) dr + r (cid:0) dη + dξ − − η ) dξ dξ + dξ (cid:1)(cid:1) . (2.30)Let us define new coordinates λ, κ, ξ ± such that dλ = − dη/ √ η , dκ = − dr/r and ξ ± = ξ ± ξ . 0 ≤ ξ ± ≤ π are periodic coordinates parametrizing a torus. The asymptotic metricis ds ≈ Gm (cid:18) dκ η + dλ + 12 η dξ − + 12 dξ (cid:19) (2.31)where η = η e −√ λ . This metric has a constant scalar curvature − /Gm . The sectionalcurvature of any coordinate plane containing ∂ ξ + vanishes due to the product form of themetric. The sectional curvatures of the remaining coordinate planes ( ∂ κ − ∂ λ , ∂ κ − ∂ ξ − , ∂ ξ − − ∂ λ )are equal to − /Gm . The scalar and sectional curvatures (of corresponding planes) of thismetric agree with the limiting values computed from the full metric on C . Near-collision geometry on S : The submersion C → S takes ( κ, λ, ξ ± ) (cid:55)→ ( λ, ξ ± ). Asthe coordinate vector fields on C are orthogonal, from (2.31) the asymptotic metric on S near the 1- 2 collision point is ds ≈ Gm (cid:18) dλ + 12 η dξ − + 12 dξ (cid:19) . (2.32)This metric has a constant scalar curvature equal to − /Gm . The sectional curvatureson the λ − ξ − coordinate 2-plane is − /Gm while it vanishes on the other two coordinate2-planes. R and C near triple collisions We argue that the triple collision configuration (which occurs at r = 0 on C or shapespace R ) is at infinite distance from other configurations with respect to the equal-mass JMmetrics (Eqs. (2.15) and (2.22)) which may be written in the form: ds = ( Gm h/r ) dr + Gm h g ij dx i dx j . (2.33) g ij is the positive (round) metric on S ( x i = ( η, ξ , ξ )) or S ( x i = ( η, ξ )) of radiusone-half: g C ij = − cos 2 η − cos 2 η and g R ij = (cid:18) η (cid:19) . (2.34) .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL § h ( ξ , η ) ≥ γ ( t ) be a curve joining a non-collision point γ ( t ) ≡ ( r , x i ) and the triple collisionpoint γ ( t ) ≡ ( r = 0 , x i ). We show that its length l ( γ ) is infinite. Since Gm hg ij is apositive matrix, l ( γ ) = (cid:90) t t dt (cid:114) Gm hr ˙ r + Gm hg ij ˙ x i ˙ x j ≥ (cid:90) t t dt (cid:114) Gm hr ˙ r . (2.35)Now using | ˙ r | ≥ − ˙ r and h ≥ 3, we get l ( γ ) ≥ −√ Gm (cid:90) t t ˙ rr dt = √ Gm (cid:90) r drr = ∞ . (2.36)In particular, a geodesic from a non-collision point to the triple collision point has infinitelength. Despite appearances, the above inequality l ( γ ) ≥ √ Gm (cid:82) r dr/r does not implythat radial curves are always geodesics. This is essentially because h along γ may be lessthan that on the corresponding radial curve. However, if ( η, ξ , ξ ) is an angular locationwhere h is minimal (locally), then the radial curve with those angular coordinates is indeeda geodesic because a small perturbation to the radial curve increases h and consequently itslength. The global minima of h ( h = 3) occur at the Lagrange configurations L , and localminima ( h = 9 / 2) are at the Euler configurations E , , indicating that radial curves at theseangular locations are geodesics. In fact, the Christoffel symbols Γ irr vanish for i = η, ξ , ξ at L , and at E , , so that radial curves γ = ( r ( t ) , x i ) satisfying ¨ r + Γ rrr ˙ r = 0 are geodesics.These radial geodesics at minima of h describe Lagrange and Euler homotheties (wherethe masses move radially inwards/outwards to/from their CM which is the center of simil-itude). These homotheties take infinite (geodesic) time to reach the triple collision. Bycontrast, the corresponding Lagrange and Euler homothety solutions to Newton’s equa-tions reach the collision point in finite time. This difference is due to an exponential time-reparametrization of geodesics relative to trajectories. In fact, if t is trajectory time and s arc-length along geodesics, then from § § σ = ds/dt = √ E + 3 Gm /r ) since h = 3. Near a triple collision (small r ), ds ≈ Gm dr /r so that s ≈ − √ Gm log(1 − t/t c ) → ∞ as t → t c = r (0) / √ Gm which is the approximate time to collision. In fact,the exact collision time t c = √ Gm (cid:16) − (cid:112) κr (0) / Gm (cid:17) /κ may be obtained by re-ducing Newton’s equations for Lagrange homotheties to the one body problem r ¨ r = − Gm whose conserved energy is κ = ˙ r − Gm /r . These homothety solutions illustrate how thegeodesic flow reformulation regularizes the original Newtonian three-body dynamics in theinverse-square potential.More generally, for unequal masses (2.10)-(2.14) give the JM metric ds = ˜ hdr /r +2 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM Figure 2.3: Gaussian curvature K (in units of 1 /Gm ) on S for equal masses and E = 0. K = 0at L , and C , , . ˜ g ij dx i dx j where˜ h = Gm m M sin η + Gm m M (cid:12)(cid:12)(cid:12) cos η − µ e iξ (cid:113) M M sin η (cid:12)(cid:12)(cid:12) + Gm m M (cid:12)(cid:12)(cid:12) cos η + µ e iξ (cid:113) M M sin η (cid:12)(cid:12)(cid:12) . (2.37)Irrespective of the masses, ˜ g ij (2.34) is positive and ˜ h has a strictly positive lower bound(e.g. Gm m M ). Thus by the same argument as above, triple collisions are at infinite dis-tance. Combining this with the corresponding results for pairwise collision points ( § C and R are geodesically complete forarbitrary masses.For non-zero energy, ds = ( E + ˜ h/r )( dr + r ˜ g ij dx i dx j ) which can be approximatedwith the zero-energy JM metrics both near binary (say, η = 0) and triple ( r = 0) collisions.If γ is a curve ending at the triple collision, l ( γ ) ≥ l (˜ γ ) where ˜ γ is a ‘tail end’ of γ lyingin a sufficiently small neighborhood of r = 0 (i.e., r (cid:28) | ˜ h/E | / which is guaranteed, say, if r (cid:28) | Gm m M /E | / ). But then, l (˜ γ ) may be estimated using the zero-energy JM metricgiving l (˜ γ ) = ∞ . Thus l ( γ ) = ∞ . A similar argument shows that curves ending at binarycollisions have infinite length. Thus we conclude that the JM metrics on C and R aregeodesically complete for arbitrary energies and masses. A geodesic through P in the direction u perturbed along v is linearly stable/unstable [see § K P ( u, v ) is positive/negative. The scalar cur-vature R at P is proportional to an average of sectional curvatures in planes through P ( § R encodes an average notion of geodesic stability. Here, we evaluate thescalar curvature R of the equal-mass zero-energy JM metric on C and its submersions to R , S and S . In each case, due to the rotation and scaling isometries, R is a function .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL η and ξ that parametrize the shape sphere. In [55] Montgomeryproves that R S ≤ C , R and S are strictly negativeand bounded below (see Fig. 2.4) indicating widespread linear instability of the geodesicdynamics. S The quotient JM metric on S (2.23) is conformal to the round (kinetic) metric on a sphereof radius 1 / ds S = Gm h ( η, ξ ) ds where ds = dη + sin η dξ . (2.38)Here the conformal factor ( h = − ( r /Gm ) × potential energy) (2.16) is a strictly positivefunction on the shape sphere with double poles at collision points. The scalar curvature of(2.38) is R S = 1 Gm h (cid:0) h + |∇ h | − h ∆ h (cid:1) , (2.39)where ∆ is the Laplacian and ∇ i h = g ij ∂ j h the gradient on S relative to the kinetic metric:∆ h = (cid:18) η ∂ h∂ξ + 2 cot 2 η ∂h∂η + ∂ h∂η (cid:19) and |∇ h | = 1sin η (cid:18) ∂h∂ξ (cid:19) + (cid:18) ∂h∂η (cid:19) . (2.40)In fact we have an explicit formula for the scalar curvature, R S = AB/C where A = 8 sin η (cid:0) (cos 2 η + 2) − η cos ξ (cid:1) ,C = 3 (cid:0) η cos 4 ξ + cos 4 η − (cid:1) and B = − η cos 8 ξ − 16 sin η cos 4 ξ (cos 4 η − η − η + 727 . (2.41)As shown in [55], R S ≤ R S also follows from (2.41): each factor in the numerator is ≥ L , ,the second at C , and the first at C ) while the denominator is strictly negative. We nowuse this to show that the scalar curvatures on center-of-mass configuration space C and itsquotients R and S are strictly negative. C The equal-mass zero-energy JM metric on C from Eq. (2.15) is ds C = (cid:0) Gm /r (cid:1) h ( η, ξ ) (cid:0) dr + r (cid:0) dη + dξ − η dξ dξ + dξ (cid:1)(cid:1) . (2.42)4 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM Figure 2.4: Scalar curvatures R on C , S and R in units of 1 /Gm . R is strictly negative andhas a global maximum at L , in all cases. It attains a global minimum at C , , on C and a localmaximum at collisions on R and S . E , , are saddles on C and global minima on R and S . The scalar curvature of this metric is expressible as R C = (cid:0) / Gm h (cid:1) (cid:0) h + |∇ h | − h ∆ h (cid:1) , (2.43)where ∆ h and ∇ h are the Laplacian and gradient with respect to the round metric on S ofradius one-half (2.40). Due to the scaling and rotation isometries, R C is in fact a functionon the shape sphere. The scalar curvatures on C (2.43) and S (2.39) are simply related: R C = 3 R S − (cid:0) / Gm h (cid:1) (cid:0) h + |∇ h | (cid:1) . (2.44)This implies R C < h is infinite, i.e., at collisions. Takingadvantage of the fact that the geometry (on S and C ) in the neighborhood of all 3 collisionpoints is the same for equal masses, it suffices to check that the second term has a strictlynegative limit at C ( η = 0). Near η = 0, h ∼ / η so that R C → − /Gm < r -independence of R C , we see that the scalar curvature is non-singularat binary and triple collisions.With a little more effort, we may obtain a non-zero upper bound for the Ricci scalar on C . Indeed, using R S ≤ h + |∇ h | ≥ ζh proved in Appendix A.2,we find R C < − ζ/ Gm where ζ = 55 / . (2.45)Numerically, we estimate the optimal value of ζ to be 8 / R and on S Recall that the equal-mass zero-energy quotient JM metrics on shape space R (2.22) and S (2.24) are ds R = (cid:0) Gm h/r (cid:1) (cid:0) dr + r (cid:0) dη + sin η dξ (cid:1)(cid:1) and ds S = Gm h (cid:0) dη + dξ − η dξ dξ + dξ (cid:1) . (2.46) .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL R R = (cid:0) h + 3 |∇ h | − h ∆ h (cid:1) / Gm h and R S = (cid:0) h + 3 |∇ h | − h ∆ h (cid:1) / Gm h . (2.47)Here ∆ h and ∇ h are as in Eq. (2.40). The scalar curvatures are related to that on S asfollows R R = 2 R S − (cid:0) h + |∇ h | (cid:1) / Gm h and R S = 2 R S − (cid:0) h + |∇ h | (cid:1) / Gm h . (2.48)As in the case of C we check that the second terms in both relations are strictly negative.This implies both the scalar curvatures are strictly negative. In fact, using the inequality12 h + |∇ h | > ζh (see Appendix A.2) we find (non-optimal) non-zero upper bounds R S , R < − ζ/ Gm where ζ = 55 / . (2.49)Moreover, we note that R C = R S − h ∆ hGm h < R S and R S = R R − h Gm h ≤ R R , (2.50)with equality at collision configurations. Recalling that on the shape sphere, the scalarcurvature vanishes at collision points (in a limiting sense) and at Lagrange points, we havethe following inequalities 0 ≥ R S > R R ≥ R S > R C . (2.51)Thus we have the remarkable result that the scalar curvatures of the JM metric on C andits quotients by scaling ( S ) and rotations ( R ) are strictly negative everywhere and alsostrictly less than that on S . So the full geodesic flow on C is in a sense more unstable thanthe corresponding flow on S .In addition to strict negativity, we may also show that the scalar curvatures are boundedbelow. For instance, from Eq. (2.39) R S can go to −∞ only when ∆ h → ∞ since h ≥ h can diverge only when sin 2 η = 0 or when one of the relevantderivatives of h diverges. From Eq. (2.16) this can happen only if η = 0 (C3) or η = π/ v i → ∞ , i.e., at collisions. However ∆ h = 66 is finite at η = π/ § R S is finite at collisions so that R S is bounded below. Thesame proof shows that scalar curvatures are bounded below on R , S and C as well. In § R on configuration space and its quotients arenegative everywhere, save at Lagrange and collision points on the shape sphere where itvanishes. However, R encodes the stability of geodesics only in an average sense. Moreprecisely, a geodesic through P in the direction u subject to a perturbation along v is6 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM linearly stable/unstable according as the sectional curvature K P ( u, v ) is positive/negative(see § u and v generalizes the Gaussian curvature to higher dimensions. It is defined as the ratio of thecurvature biquadratic r = g ( R ( u, v ) v, u ) to the square of the area Ar( u, v ) = g ( u, u ) g ( v, v ) − g ( u, v ) g ( v, u ) of the parallelogram spanned by u and v . Here g ( u, v ) is the Riemannian innerproduct and R ( u, v ) = [ ∇ u , ∇ v ] − ∇ [ u,v ] the curvature tensor with components R ( e i , e j ) e k = R lkij e l in any basis for vector fields. Furthermore, if e , . . . , e n are an orthonormal basisfor the tangent space at P , then the scalar curvature R = (cid:80) i (cid:54) = j K ( e i , e j ) is the sum ofsectional curvatures in (cid:0) n (cid:1) planes through P . It may also be regarded as an average of thecurvature biquadratic R = (cid:82)(cid:82) r ( u, v ) dµ g ( u ) dµ g ( v ) where dµ g ( u ) = exp ( − u i u j g ij / du isthe gaussian measure on tangent vectors with mean zero and covariance g ij [67]. Thus R provides an averaged notion of stability. To get a more precise measure of linear stabilityof geodesics we find the sectional curvatures in various (coordinate) tangent 2-planes of theconfiguration space and its quotients. On account of the isometries, these sectional curvaturesare functions only of η and ξ [explicit expressions are omitted due to their length]. Unlikescalar curvatures which were shown to be non-positive, we find planes in which sectionalcurvatures are non-positive as well as planes where they can have either sign.O’Neill’s theorem allows us to determine or bound certain sectional curvatures on thecenter-of-mass configuration space C in terms of the more easily determined curvatures onits quotients. Roughly, the sectional curvature of a horizontal two-plane increases under aRiemannian submersion. Suppose f : ( M, g ) → ( N, ˜ g ) is a Riemannian submersion. ThenO’Neill’s theorem [63] states that the sectional curvature in any horizontal 2-plane at m ∈ M is less than or equal to that on the corresponding 2-plane at f ( m ) ∈ N : K N ( df ( X ) , df ( Y )) = K M ( X, Y ) + 34 | [ X, Y ] V | Ar( X, Y ) . (2.52)Here X and Y are horizontal fields on M spanning a non-degenerate 2-plane (Ar( X, Y ) (cid:54) =0) and [ X, Y ] V is the vertical projection of their Lie bracket. In particular, the sectionalcurvatures are equal everywhere if X and Y are coordinate vector fields.We consider sectional curvatures in 6 interesting 2 planes on C which are horizontalwith respect to submersions to R and S . Under the submersion from C to R ( § ∂ r , ∂ η and ∂ ξ ≡ cos 2 η∂ ξ + ∂ ξ map respectively to ∂ r , ∂ η and ∂ ξ defining three pairs of corresponding 2 -planes. Since [ ∂ r , ∂ η ] and [ ∂ r , ∂ ξ ] vanish,we have K C ( ∂ r , ∂ η ) = K R ( ∂ r , ∂ η ) and K C ( ∂ r , ∂ ξ ) = K R ( ∂ r , ∂ ξ ). Fig. 2.5 shows that K C ( ∂ r , ∂ η ) is mostly negative, though it is not continuous at E , C and C . On theother hand K C ( ∂ r , ∂ ξ ) is largely negative except in a neighborhood of C . Finally, as[ ∂ ξ , ∂ η ] V = − η∂ ξ (cid:54) = 0, we have K C ( ∂ η , ∂ ξ ) < K R ( ∂ η , ∂ ξ ) with equality at collisions.Moreover the submersion from R → S ( § K R ( ∂ η , ∂ ξ ) coincides with K S ( ∂ η , ∂ ξ ) which vanishes at Lagrange and collision points and is strictly negative elsewhere(see § K C ( ∂ η , ∂ ξ ) vanishes at collision points and is strictly negative everywhereelse (see Fig. 2.5). In particular, Lagrange points are more unstable on the configurationspace C than on the shape sphere. .2. PLANAR THREE–BODY PROBLEM WITH INVERSE–SQUARE POTENTIAL (a) (b) (c) Figure 2.5: Sectional curvatures on horizontal 2-planes of submersion from C to R in units of1 /Gm . (a) K C ( ∂ r , ∂ η ) = K R ( ∂ r , ∂ η ) ≤ E . K = − C and K = − / L , . K → , − C , are approached holding η or ξ fixed. (b) K C ( ∂ r , ∂ ξ ) = K R ( ∂ r , ∂ ξ ) is negative except in neighborhoods of C and E . K = 0 at its minimum C ( η = 0) and K = − / L , . K → − C , ( η = π/ , ξ = 0 , π/ 2) along η or ξ constant. (c) K C ( ∂ η , ∂ ξ ) ≤ K R ( ∂ η , ∂ ξ ). K C ( ∂ η , ∂ ξ ) = 0 atglobal maxima C , , and is negative elsewhere. K = − L , . Under the submersion from C to S ( § ∂ η , ∂ ξ and ∂ ξ map respectively to ∂ η , ∂ ξ and ∂ ξ . The sectional curvatures on corresponding pairsof 2-planes are equal, e.g. K C ( ∂ η , ∂ ξ ) = K S ( ∂ η , ∂ ξ ). As shown in Fig. 2.6, K C ( ∂ η , ∂ ξ )is negative everywhere except in a neighborhood of E where it can have either sign. Thequalitative behavior of the other two sectional curvatures K C ( ∂ ξ , ∂ ξ ) and K C ( ∂ ξ , ∂ η ) issimilar to that of K C ( ∂ r , ∂ ξ ) and K C ( ∂ r , ∂ η ) discussed above. The approximate symmetryunder ∂ ξ ↔ ∂ r is not entirely surprising given that ∂ ξ and ∂ r are vertical vectors in thesubmersions to R and S respectively.The remaining two coordinate 2-planes on C are not horizontal under either submersion.We find that K C ( ∂ r , ∂ ξ ) is negative everywhere except at L , and K C ( ∂ r , ∂ ξ ) is negativeexcept around E , . In this section we use the stability tensor (which provides a criterion for linear geodesicstability) to discuss the stability of Lagrange rotational and homothety solutions. We endwith a remark on linear stability of trajectories and geodesics. Consider the n -dimensionalconfiguration manifold M with metric g . The geodesic deviation equation (GDE) for theevolution of the separating vector (Jacobi field) y ( t ) between a geodesic x ( t ) and a neigh-boring geodesic is [63] ∇ x y = R ( ˙ x, y ) ˙ x = − R ( y, ˙ x ) ˙ x. (2.53)We expand the Jacobi field y = c k ( t ) e k ( t ) in any basis e i ( t ) that is parallel transported alongthe geodesic i.e. ∇ ˙ x e k = 0 [ e i (0) could be taken as coordinate vector fields at x (0)]. Taking8 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM (a) (b) (c) Figure 2.6: Sectional curvatures on horizontal 2-planes of submersion from C to S in units of1 /Gm . (a) K C ( ∂ η , ∂ ξ ) = K S ( ∂ η , ∂ ξ ) > E and negative elsewhere. K = − C . K = − L , . K → − / C , along constant η or ξ . (b) K C ( ∂ η , ∂ ξ ) = K S ( ∂ η , ∂ ξ ) > E and is negative elsewhere. K = − C and K = − / L , . K → − C , holding η or ξ fixed. (c) K C ( ∂ ξ , ∂ ξ ) = K S ( ∂ ξ , ∂ ξ ) > C and E and negative elsewhere. K = 0 at its local minimum C . K = − / L , . K → − C , while holding η or ξ fixed. the inner product of the GDE with e m and contracting with g im , we get ¨ c i = − S ij c j , wherethe ‘stability tensor’ S ik = R ijkl ˙ x j ˙ x l . As S is real symmetric, its eigenvectors f i can be chosento form an orthonormal basis for T x M . Writing y = d m f m , the GDE becomes ¨ d m = − κ m d m (no sum on m ) where κ m is the eigenvalue of S corresponding to the eigenvector f m .The eigenvalues of S (say at t = 0) control the initial evolution of the Jacobi fields inthe corresponding eigendirections. Since κ m = (Area (cid:104) f k , ˙ x (cid:105) ) K ( f m , ˙ x ) ( § κ or K imply local stability (instability) for the initial evolution. We note thatcalculating S and its eigenvalues at a given instant (say t = 0) requires no knowledge of thetime evolution of e i ( t ). So we may simply use the coordinate vector fields as the basis. Noticethat the tangent vector to the geodesic ˙ x is always an eigendirection of S with eigenvaluezero. Consider the Lagrange rotational solutions where three equal masses ( m i = m ) rotate atangular speed ω = (cid:112) Gm/a around their CM at the vertices of an equilateral trian-gle of side a . The rotational trajectory on C in r, η, ξ , coordinates is given by x ( t ) =( a/ √ m, π/ , ωt, ± π/ 4) with velocity vector ω∂ ξ . Note that trajectory and geodesic timesare proportional since σ = ds/dt = ( E − V ) / √T with V ( r, η, ξ ) and T constant along x ( t ). The stability tensor along the geodesic, S = ω diag(1 , − / , , − / 2) is diagonal inthe coordinate basis r, η, ξ , ξ . As always, ˙ x is a zero-mode. A perturbation along ∂ r islinearly stable while those directed along ∂ η or ∂ ξ are linearly unstable. Note that Routh’scriterion 27( m m + m m + m m ) < M [70] predicts that Lagrange rotational solutions .3. PLANAR THREE–BODY PROBLEM WITH NEWTONIAN POTENTIAL For equal masses, a Lagrange homothety solution is one where the masses move radially(towards/away from their CM) while being at the vertices of equilateral triangles. Thegeodesic in Hopf coordinates takes the form ( r ( t ) , η = π/ , ξ , ξ = ± π/ 4) where ξ isarbitrary and independent of time. Though an explicit expression is not needed here, r ( t )is the solution of ¨ r + Γ rrr ˙ r = 0 where Γ rrr = − Gm / ( Er + 3 Gm r ) for the inverse-squarepotential. The stability tensor is diagonal: S = 6 Gm ˙ r (3 Gm r + Er ) diag (cid:0) , − Gm − Er , − Er , − Gm − Er (cid:1) . (2.54)For a given r and positive energy, perturbations along ∂ ξ , and ∂ η are unstable while theyare stable when − Gm /r < E < − Gm / r . For intermediate (negative) energies, ∂ η and ∂ ξ are unstable directions while ∂ ξ is stable. For the Newtonian potential, we havesimilar conclusions following from the corresponding stability tensor: S = 3 Gm / ˙ r r (3 Gm / + Er ) diag (cid:0) , − Gm / − Er, − Er, − Gm / − Er (cid:1) . (2.55)We end this section with a cautionary remark. For a system whose trajectories can beregarded as geodesics of the JM metric, linear stability of geodesics may not coincide withlinear stability of corresponding trajectories. This may be due to the reparametrization oftime (see § k . Here thecurvature of the JM metric (see § R = 2 Ek/T where T is the kinetic energy. Thus forpositive k , geodesics are always linearly stable while for negative k they are stable/unstableaccording as energy is negative/positive. By contrast, linearizing the EOM ¨ δx i = − ( k/m ) δx i shows that trajectories are linearly stable for positive k and linearly unstable for negative k . This (possibly atypical) example illustrates the fact that geodesic stability does notnecessarily imply stability of trajectories. In analogy with our geometric treatment of the planar motion of three masses subject toinverse-square potentials, we briefly discuss the gravitational analogue with Newtonian po-0 CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM tentials. As before, the translation invariance of the Lagrangian L = 12 (cid:88) i =1 , , m i ˙ x i − (cid:88) i 2. Evidently, both the scalar curvatures vanish in the limit r → ∞ of large .3. PLANAR THREE–BODY PROBLEM WITH NEWTONIAN POTENTIAL (a) (b) Figure 2.7: Ricci scalar R for zero energy and equal masses on C and R for the Newtonianpotential (in units of 1 /Gm / r ). R on C is strictly negative while that on R can have eithersign. moment of inertia I CM = r ; they are plotted in Fig. 2.7. Numerically, we find that forany fixed r , R C is strictly negative and reaches its global maximum − / (2 Gm / r ) at theLagrange configurations L , , while R R has a positive global maximum 1 / (2 Gm / r ) atthe same locations. Note that R R = 2 R C / U + |∇ U | ) / (2 Gm / rU ). As arguedin Eq. (2.44), the second term is strictly positive and vanishes only when r → ∞ . Usingthe negativity of R C , it follows that R R > R C with ( R R − R C ) attaining its minimum2 / ( Gm / r ) at L , . Thus in a sense, the geodesic dynamics on C is more linearly unstablethan on shape space. Like the Ricci scalars, sectional curvatures on coordinate 2-planes are(1 /r ) × a function of η and ξ . We find that sectional curvatures are largely negative andoften go to ±∞ at collision points (see Eq. (2.63)). Unlike for the inverse-square potential, the scalar curvatures on C and R (2.61) diverge atbinary and triple collisions. To examine the geometry near pairwise collisions of equal masses,it suffices to study the geometry near C ( η = 0, r (cid:54) = 0, ξ , arbitrary) which representsa collision of m and m . We do so by retaining only those terms in the expansion of thezero-energy metrics around η = 0: ds C ≈ (cid:18) Gm / √ ηr (cid:19) (cid:0) dr + r (cid:0) dη + dξ − − η ) dξ dξ + dξ (cid:1)(cid:1) and ds R ≈ (cid:18) Gm / r (cid:19) (cid:32) √ η + 2 (cid:114) (cid:33) (cid:0) dr + r (cid:0) dη + 4 η dξ (cid:1)(cid:1) , (2.62)that are necessary to arrive at the following curvatures to leading order in η :on C : R = − (cid:37) and K ( ∂ η , ∂ r,ξ , ) = 2 K ( ∂ r , ∂ ξ , ) = − K ( ∂ ξ , ∂ ξ ) = − (cid:37) CHAPTER 2. INSTABILITIES IN THE PLANAR THREE–BODY PROBLEM on R : R = − (cid:37) , K ( ∂ η , ∂ r ) = − K ( ∂ r , ∂ ξ ) = − (cid:37) and K ( ∂ η , ∂ ξ ) = − (cid:112) / Gm / (2.63)where (cid:37) = √ Gm / ηr . The curvature singularity at η = 0 is evident in the simple poles inthe Ricci scalars and all but one of the sectional curvatures in coordinate planes.We use the near-collision JM metric of Eq. (2.62) to show that a pairwise collision pointlies at finite geodesic distance from another point in its neighborhood. Thus, unlike forthe inverse-square potential, the geodesic reformulation does not regularize the gravitationalthree-body problem. Consider a point P near η = 0 with coordinates ( r, η , ξ , ξ ). Weestimate its distance to the collision point C ( r, , ξ , ξ ). To do so, we consider a curve γ of constant r , ξ and ξ running from P to C parametrized by η ≥ η ≥ 0. We will showthat γ has finite length so that the geodesic distance to C must be finite. In fact, from(2.62): Length( γ ) = (cid:90) η (cid:115) Grm / √ dη √ η = − (cid:115) Grm / √ √ η < ∞ . (2.64)Furthermore, the image of γ under the Riemannian submersion to shape space R is acurve of even shorter length ending at a collision point. Thus geodesics on C and R canreach binary collisions in finite time, where the scalar curvature is singular. It is thereforeinteresting to study regularizations of collisions in the three body problem and their geometricinterpretation. hapter 3Instabilities, chaos and ergodicity inthe classical three-rotor problem In this chapter, we investigate periodic orbits, instabilities and onset of chaos in the systemof three coupled rotors. Furthermore, we investigate ergodicity, mixing and recurrence timestatistics in a band of energies. This chapter is based on [42], [43] and [44]. We study a periodic chain of three identical rotors of mass m interacting via attractive cosinepotentials. The Lagrangian is L = (cid:88) i =1 (cid:26) mr ˙ θ i − g [1 − cos ( θ i − θ i +1 )] (cid:27) (3.1)with θ ≡ θ . Here, θ i are 2 π -periodic coordinates on a circle of radius r . Though weonly have nearest neighbor interactions, each pair interacts as there are only three rotors.We consider the ‘ferromagnetic’ case where the coupling g > i = 1, 2 and 3 (with θ ≡ θ and θ ≡ θ ) are mr ¨ θ i = g sin( θ i − − θ i ) − g sin( θ i − θ i +1 ) . (3.2)This is a system with three degrees of freedom, the configuration space is a 3-torus 0 ≤ θ i ≤ π . The conjugate angular momenta are π i = mr ˙ θ i and the Hamiltonian is H = (cid:88) i =1 (cid:26) π i mr + g [1 − cos ( θ i − θ i +1 )] (cid:27) . (3.3)334 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM Hamilton’s equations˙ θ i = π i mr and ˙ π i = g [sin( θ i − − θ i ) − sin( θ i − θ i +1 )] (3.4)define a smooth Hamiltonian vector field on the 6d phase space of the three-rotor problem.The additive constant in H is chosen so that the minimal value of energy is zero. This systemhas three independent dimensionful physical parameters m , r and g that can be scaled to oneby a choice of units. However, once such a choice of units has been made, all other physicalquantities (such as (cid:126) ) have definite numerical values. This circumstance is similar to that inthe Toda model [31]. As discussed in Appendix B.1, the quantum n -rotor problem, whichmodels a chain of Josephson junctions, also arises by Wick-rotating a partial continuum limitof the XY model on a lattice with nearest neighbor ferromagnetic coupling J , n horizontalsites and horizontal and vertical spacings a and b (B.7). The above parameters are relatedto those of the Wick-rotated XY model via m = J/c , r = (cid:112) Lb /a and g = J L/a where L = na and c is a speed associated to the Wick rotation to time.The Hamiltonian vector field (3.4) is non-singular everywhere on the phase space. Inparticular, particles may pass through one another without encountering collisional singular-ities. Though the phase space is not compact, the constant energy ( H = E ) hypersurfacesare compact 5d submanifolds without boundaries. Indeed, 0 ≤ θ i ≤ π are periodic coordi-nates on the compact configuration space T . Moreover, the potential energy is non-negativeso that π i ≤ mr E . Thus, the angular momenta too have finite ranges. Consequently, wecannot have ‘non-collisional singularities’ where the (angular) momentum or position divergesin finite time. Solutions to the initial value problem (IVP) are therefore expected to existand be unique for all time.Alternatively, the Hamiltonian vector field is globally Lipschitz since it is everywheredifferentiable and its differential bounded in magnitude on account of energy conservation.This means that there is a common Lipschitz constant on the energy hypersurface, so that aunique solution to the IVP is guaranteed to exist for 0 ≤ t ≤ t where t > t ≤ t ≤ t and thus can be prolonged indefinitely in time for any IC, implying global existence anduniqueness [22].In § E > . g , this is expected on account of compactness and lack of boundaryof the energetically allowed Hill region. For E < . g , though the trajectories can (in finitetime) reach the Hill boundary, they simply turn around. Examples of such trajectories areprovided by the ϕ = 0 pendulum solutions described in § .1. THREE COUPLED CLASSICAL ROTORS It is convenient to define the center of mass (CM) and relative angles ϕ = θ + θ + θ , ϕ = θ − θ and ϕ = θ − θ (3.5)or equivalently, θ = ϕ + 2 ϕ ϕ , θ = ϕ − ϕ ϕ θ = ϕ − ϕ − ϕ . (3.6)As a consequence of the 2 π -periodicity of the θ s, ϕ is 2 π -periodic while ϕ , are 6 π -periodic.However, the cuboid ( 0 ≤ ϕ ≤ π , 0 ≤ ϕ , ≤ π ) is a nine-fold cover of the fundamentalcuboid 0 ≤ θ , , ≤ π . In fact, since the configurations ( ϕ , ϕ − π, ϕ ), ( ϕ , ϕ , ϕ + 2 π )and ( ϕ + 2 π/ , ϕ , ϕ ) are physically identical, we may restrict ϕ , to lie in [0 , π ]. Here,the ϕ i are not quite periodic coordinates on T ≡ [0 , π ] . Rather, when ϕ (cid:55)→ ϕ ± π or ϕ (cid:55)→ ϕ ∓ π , the CM variable ϕ (cid:55)→ ϕ ± π/ 3. In these coordinates, the Lagrangianbecomes L = T − V where T = 32 mr ˙ ϕ + 13 mr (cid:2) ˙ ϕ + ˙ ϕ + ˙ ϕ ˙ ϕ (cid:3) and V = g [3 − cos ϕ − cos ϕ − cos( ϕ + ϕ )] , (3.7)with the equations of motion (EOM) 3 mr ¨ ϕ = 0, mr (2 ¨ ϕ + ¨ ϕ ) = − g [sin ϕ + sin( ϕ + ϕ )] and 1 ↔ . (3.8)The evolution equations for ϕ (and ϕ with 1 ↔ 2) may be rewritten as mr ¨ ϕ = − g [2 sin ϕ − sin ϕ + sin( ϕ + ϕ )] . (3.9)Notice that when written this way, the ‘force’ on the RHS isn’t the gradient of any potential,as the equality of mixed partials would be violated. The (angular) momenta conjugate to ϕ , , are p = 3 mr ˙ ϕ , p = mr ϕ + ˙ ϕ ) and p = mr ϕ + 2 ˙ ϕ ) . (3.10)The remaining three EOM on phase space are ˙ p = 0 (conserved due to rotation invariance),˙ p = − g [sin ϕ + sin( ϕ + ϕ )] and ˙ p = − g [sin ϕ + sin( ϕ + ϕ )] . (3.11)The EOM admit a conserved energy which is a sum of CM, relative kinetic and potentialenergies: E = 32 mr ˙ ϕ + 13 mr (cid:2) ˙ ϕ + ˙ ϕ + ˙ ϕ ˙ ϕ (cid:3) + V ( ϕ , ϕ ) . (3.12)The above EOM are Hamilton’s equations ˙ f = { f, H } for canonical Poisson brackets (PBs) { ϕ i , p j } = δ ij with the Hamiltonian H = p mr + p + p − p p mr + V ( ϕ , ϕ ) . (3.13)6 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM We define the Jacobi coordinates for the three-rotor problem to be ϕ (3.5) and ϕ + = ( ϕ + ϕ ) / θ − θ ) / ϕ − = ( ϕ − ϕ ) / θ + θ ) / − θ . (3.14)Up to a change in order, these are analogous to the Jacobi vectors of the three-body problem(see Fig. 2.1): ϕ is the center of mass of the three rotors, 2 ϕ + is the angle of the first rotorrelative to the third and − ϕ − is the angle of the second rotor with respect to the centerof mass of the first and the third rotors. Unlike in the CM and relative coordinates and asin the three-body problem, the kinetic energy as a quadratic form in velocities is diagonal.Indeed, L = T − V where T = 32 mr ˙ ϕ + mr ˙ ϕ + 13 mr ˙ ϕ − and V = g (3 − ϕ − cos ϕ + − cos 2 ϕ + ) . (3.15)The conjugate momenta p and p ± = p ± p are proportional to the velocities and the EOMare ˙ p = 0 , ˙ p + = − g sin ϕ + (cos ϕ − + 2 cos ϕ + ) and ˙ p − = − g cos ϕ + sin ϕ − . (3.16)The fundamental domain which was the cube 0 ≤ ϕ , , ≤ π now becomes the cuboid(0 ≤ ϕ ≤ π , 0 ≤ ϕ + ≤ π , 0 ≤ ϕ − ≤ π ). As before, though ϕ ± are periodic coordinateson a 2-torus, ϕ , ± are not quite periodic coordinates on T . The transformation of the CMvariable ϕ under 2 π -shifts of ϕ , discussed above may be reformulated as follows. Whencrossing the segments ϕ + + ϕ − = 2 π from left to right or ϕ + − ϕ − = 0 from right to left, ϕ increases by 2 π/ ϕ (cid:55)→ ϕ − π/ ϕ - ϕ torus The dynamics of ϕ and ϕ (or equivalently that of ϕ ± ) decouples from that of the CMcoordinate ϕ . The former may be regarded as periodic coordinates on the 2-torus [0 , π ] × [0 , π ]. On the other hand, ϕ , which may be regarded as a fibre coordinate over the ϕ , base torus, evolves according to ϕ = p t mr + ϕ (0) + 2 π n − n ) mod 2 π. (3.17)Here, n , are the ‘greatest integer winding numbers’ of the trajectory around the cyclesof the base torus. If a trajectory goes continuously from ϕ i , to ϕ f , (regarded as realrather than modulo 2 π ), then the greatest integer winding numbers are defined as n , =[( ϕ f , − ϕ i , ) / π ].Consequently, we may restrict attention to the dynamics of ϕ and ϕ . The equations ofmotion on the corresponding 4d phase space (the cotangent bundle of the 2-torus) are˙ ϕ = (2 p − p ) /mr , ˙ p = − g [sin ϕ + sin( ϕ + ϕ )] (3.18) .2. DYNAMICS ON THE ϕ – ϕ TORUS (a) Contours of V . (b) Ground state G. (c) Diagonal states D.(d) Triangle states T. Figure 3.1: (a) Potential energy V in units of g on the ϕ - ϕ configuration torus with its extrema(locations of static solutions G, D and T) indicated. The contours also encode changes in topologyof the Hill region ( V ≤ E ) when E crosses E G = 0 , E D = 4 g and E T = 4 . g . (b, c, d) Uniformlyrotating three-rotor solutions obtained from G, D and T. Here, i, j and k denote any permutationof the numerals 1 , 2 and 3 . (b) and (d) are the simplest examples of choreographies discussed in § and 1 ↔ 2. These equations define a singularity-free vector field on the phase space. Theyfollow from the canonical PBs with Hamiltonian given by the relative energy H rel = p + p − p p mr + V ( ϕ , ϕ ) . (3.19)These equations and Hamiltonian are reminiscent of those of the planar double pendulumwith the Hamiltonian H dp = p − c p p + 2 p ml (2 − c ) − mgl (2 cos θ + cos θ ) (3.20)where θ , are the angles between the upper and the lower rods (each of length l ) and thevertical and c = cos( θ − θ ). Static solutions for the relative motion correspond to zeros of the vector field where the forcecomponents in (3.18) vanish: p = p = 0 andsin ϕ + sin( ϕ + ϕ ) = sin ϕ + sin( ϕ + ϕ ) = 0 . (3.21)8 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM In particular, we must have ϕ = ϕ or ϕ = π − ϕ . When ϕ = ϕ , the force componentsare both equal to sin ϕ (1 + 2 cos ϕ ) which vanishes at the following configurations:( ϕ , ϕ ) = (0 , , ( π, π ) and ( ± π/ , ± π/ . (3.22)On the other hand, if ϕ = π − ϕ , we must have sin ϕ = 0 leading to two more staticconfigurations (0 , π ) and ( π, E = 0 : G (0 , ,E = 4 g : D ( π, π ) , D ( π, , D (0 , π )and E = 9 g/ T , ( ± π/ , ± π/ . (3.23)Below, we clarify their physical meaning by viewing them as uniformly rotating three bodyconfigurations. If we include the uniform rotation of the CM angle ( ˙ ϕ = Ω is arbitrary), these six so-lutions correspond to the following uniformly rotating rigid configurations of three-rotors(see Fig. 3.1): (a) the ferromagnetic ground state G where the three particles coalesce( θ = θ = θ ), (b) the three ‘diagonal’ ‘anti ferromagnetic N´eel’ states D where two particlescoincide and the third is diametrically opposite ( θ = θ = θ + π and cyclic permutationsthereof) and (c) the two ‘triangle’ ‘spin wave’ states T where the three bodies are equallyseparated ( θ = θ + 2 π/ θ + 4 π/ θ ↔ θ ). The linearization of the EOM (3.9) for perturbations to G, D and T ( ϕ , = ¯ ϕ , + δϕ , ( t ))take the form mr d dt (cid:18) δϕ δϕ (cid:19) = − gA (cid:18) δϕ δϕ (cid:19) where A G = 3 I,A D (0 ,π ) = (cid:18) − − (cid:19) , A D ( π, = (cid:18) − − 20 1 (cid:19) ,A D ( π,π ) = (cid:18) − − (cid:19) and A T = − I/ . (3.24)Here I is the 2 × ω = (cid:112) g/mr . The saddles D have one stable direction withfrequency ω / √ ω . On the other hand,both eigendirections around T are unstable with growth rate ω / √ .2. DYNAMICS ON THE ϕ – ϕ TORUS Adding 2 discs E > 4.5g4g< E < 4.5g0< E < 4g Adding 3 1-cells (a) (b) Figure 3.2: (a) Topology of Hill region of configuration space ( V ( ϕ , ϕ ) ≤ E ) showing transitionsat E = 4 g and 4 . g as implied by Morse theory (see § E = 4 g is notquite a manifold; its boundary consists of 3 non-contractible closed curves on the torus meeting atthe D configurations. The Hill region of possible motions H E at energy E is the subset V ( ϕ , ϕ ) ≤ E of the ϕ - ϕ configuration torus. The topology of the Hill region for various energies can be read-off fromFig. 3.1a. For instance, for 0 < E < g , H E is a disc while it is the whole torus for E > . g .For 4 g < E < . g , it has the topology of a torus with a pair of discs (around T and T )excised (see also Fig. 3.7). These changes in topology are confirmed by Morse theory [52] ifwe treat V as a real-valued Morse function, since its critical points are non-degenerate (non-singular Hessian). In fact, the critical points of V are located at G (minimum with index0), D , , (saddles with indices 1) and T , (maxima with indices 2). Thus, the topology of H E can change only at the critical values E G = 0 , E D = 4 g and E T = 4 . g (see Fig. 3.2a).The topological transition from H E< g (disc) to H g The linear equations of motion for ϕ and ϕ decouple, mr ¨ ϕ = − gϕ and mr ¨ ϕ = − gϕ (3.26)leading to the separately conserved normal mode energies E , = (cid:0) mr ˙ ϕ , + 3 gϕ , (cid:1) / 2. Theequality of frequencies implies that any pair of independent linear combinations of ϕ and ϕ are also normal modes. Of particular significance are the Jacobi-like variables ϕ ± =( ϕ ± ϕ ) / L low = mr ˙ ϕ − gϕ + mr ˙ ϕ − / − gϕ − . (3.27)Though (3.26) are simply the EOM for a pair of decoupled oscillators, the Lagrangian andPoisson brackets {· , ·} inherited from the non-linear theory are different from the standardones. With conjugate momenta p , = ( mr / ϕ , + ˙ ϕ , ), the Hamiltonian correspondingto (3.25) is H low = p − p p + p mr + g (cid:0) ϕ + ϕ + ϕ ϕ (cid:1) . (3.28)Note that p , differ from the standard momenta p s1 , = mr ˙ ϕ , whose PBs are now non-canonical, { ϕ i , p s j } = − δ ij . H low and the normal mode energies H , = (2 p , − p , ) / mr + 3 gϕ , / dH , dH and dH are generically linearly independent ( dH ∧ dH ∧ dH (cid:54)≡ L z = mr ( ϕ ˙ ϕ − ϕ ˙ ϕ ) corresponding to the rotation invariance of the decoupled oscillators in (3.26). It turnsout that H low may be expressed as H low = 23 (cid:104) H + H + (cid:112) H H − (3 g/ mr ) L z (cid:105) . (3.30)The low energy phase trajectories lie on the common level curves of H low , H and H .Though H and H are conserved energies of the normal modes, they do not Poisson com-mute. In fact, the Poisson algebra of conserved quantities is { H , , H low } = { L z , H low } = 0, { H , H } = − gL z /mr and { L z , H , } = ± H low − H , − H , ) . (3.31)It is also noteworthy that the integrals H + H and H H − gL z / (4 mr ) are in involution. .3. REDUCTIONS TO ONE DEGREE OF FREEDOM CM θ i θ k θ j (a) Pendula θ j θ k θ i (b) Isosceles ‘breathers’ Figure 3.3: In pendula, θ i and θ j form a molecule that along with θ k oscillates about their commonCM. In breathers, θ i is at rest at the CM with θ j and θ k oscillating symmetrically about the CM.Here, i, j and k denote any permutation of the numerals 1 , 2 and 3. Recall that the Euler and Lagrange solutions of the planar three-body problem arise througha reduction to the two body Kepler problem. We find an analogue of this construction forthree rotors, where pendulum-like systems play the role of the Kepler problem. We find twosuch families of periodic orbits, the pendula and isosceles breathers (see Fig. 3.3). They existat all energies and go from librational to rotational motion as E increases. They turn out tohave remarkable stability properties which we deduce via their monodromy matrices. We seek solutions where one pair of rotors form a ‘bound state’ with their angular separationremaining constant in time. We show that consistency requires this separation to vanish,so that the two behave like a single rotor and the equations reduce to that of a two-rotorproblem. There are three such families of ‘pendulum’ solutions depending on which pair isbound together (see Fig. 3.3a). For definiteness, we suppose that the first two particles havea fixed separation ζ ( θ = θ + ζ or ϕ = ζ ). Putting this in (3.9), we get a consistencycondition and an evolution equation for ϕ :2 sin ζ − sin ϕ + sin( ζ + ϕ ) = 0 and mr ¨ ϕ = − g [2 sin ϕ − sin ζ + sin( ζ + ϕ )] . (3.32)The consistency condition is satisfied only when the separation ζ = 0 , i.e., rotors 1 and 2must coincide so that ϕ = 0 and ˙ ϕ = 0 (or p = 2 p ) at all times (the other two familiesare defined by ϕ = ˙ ϕ = 0 and ϕ + ϕ = ˙ ϕ + ˙ ϕ = 0). The evolution equation for ϕ reduces to that for a simple pendulum: mr ¨ ϕ = − g sin ϕ with E = mr ˙ ϕ g (1 − cos ϕ ) (3.33)2 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM being the conserved energy. The periodic solutions are either librational (for 0 ≤ E < g )or rotational (for E > g ) and may be expressed in terms of the Jacobi elliptic function sn:¯ ϕ ( t ) = (cid:40) k sn( ω t, k )) for 0 ≤ E ≤ g, ω t/κ, κ )) for E ≥ g. (3.34)Here, ω = (cid:112) g/mr and the elliptic modulus k = (cid:112) E/ g with κ = 1 /k . Thus 0 ≤ k < ≤ κ < τ lib = 4 K ( k ) /ω and τ rot = 2 κK ( κ ) /ω , where K is the complete elliptic integral of the first kind. As E → g ± ,the period diverges and we have the separatrix ¯ ϕ ( t ) = 2 arcsin(tanh( ω t )). The conditions ϕ = 0 and p = 2 p define a 2d ‘pendulum submanifold’ of the 4d phase space foliated bythe above pendulum orbits. Upon including the CM motion of ϕ , each of these periodicsolutions may be promoted to a quasi-periodic orbit of the three-rotor problem. There is atwo-parameter family of such orbits, labelled for instance, by the relative energy E and theCM angular momentum p . Introducing the dimensionless variables˜ p , = p , / (cid:112) mr g and ˜ t = t (cid:112) g/mr , (3.35)the equations for small perturbations ϕ = δϕ , ϕ = ¯ ϕ + δϕ and p , = ¯ p , + δp , (3.36)to the above pendulum solutions (3.34) to (3.18) are d d ˜ t (cid:18) δϕ δϕ (cid:19) = − (cid:18) ϕ ϕ − ϕ (cid:19) (cid:18) δϕ δϕ (cid:19) . (3.37)This is a pair of coupled Lam´e-type equations since ¯ ϕ is an elliptic function. The analogousequation in the 2d anharmonic oscillator reduces to a single Lam´e equation [6, 82]. Ourcase is a bit more involved and we will resort to a numerical approach here. To do so, it isconvenient to consider the first order formulation dd ˜ t δϕ δϕ δ ˜ p δ ˜ p = − − − 21 + cos ¯ ϕ cos ¯ ϕ ϕ ϕ δϕ δϕ δ ˜ p δ ˜ p . (3.38)Since m, g and r have been scaled out, there is no loss of generality in working in unitswhere m = g = r = 1 , as we do in the rest of this section. The solution is ψ ( t ) = U ( t, ψ (0) where the real time-evolution matrix is given by a time-ordered exponential U ( t, 0) = T exp { (cid:82) t A ( t ) dt } where A ( t ) is the coefficient matrix in (3.38) and T denotes .3. REDUCTIONS TO ONE DEGREE OF FREEDOM A ( t ) implies det U ( t, 0) = 1 and preservation of phasevolume. Though A ( t ) is τ -periodic, ψ ( t + τ ) = M ( τ ) ψ ( t ) where the monodromy matrix M ( τ ) = U ( t + τ, t ) is independent of t . Thus, ψ ( t + nτ ) = M n ψ ( t ) for n = 1 , , . . . , sothat the long-time behavior of the perturbed solution may be determined by studying thespectrum of M . In fact, the eigenvalues λ of M may be related to the Lyapunov exponentsassociated to the pendulum solutions µ = lim t →∞ t ln | ψ ( t ) || ψ (0) | via µ = log | λ | τ . (3.39)Since ours is a Hamiltonian system with 2 degrees of freedom, two of the eigenvalues of M must equal one and the other two must be reciprocals [34]. On account of the reality of M ,the latter two ( λ , λ ) must be of the form ( e iφ , e − iφ ) or ( λ, /λ ) where φ and λ are real. Itfollows that two of the Lyapunov exponents must vanish while the other two must add up tozero. The stability of the pendulum orbit is governed by the stability index σ = tr M − | σ | ≤ e ± iφ and instability if | σ | = | λ + 1 /λ | > A ( t ) becomes time-independent andsimilar to 2 πi × diag(1 , , − , − M = exp( Aτ ) is the 4 × I .Thus G is stable and small perturbations around it are periodic with period τ = 2 π/ω , aswe know from Eq. (3.24). For E > 0, we evaluate M numerically. We find it more efficientto regard M as the fundamental matrix solution to ˙ ψ = A ( t ) ψ rather than as a path orderedexponential or as a product of infinitesimal time-evolution matrices. Remarkably, as discussedbelow, we find that while the system is stable for low energies 0 ≤ E ≤ E (cid:96) ≈ . 99 and highenergies E ≥ E r ≈ . 60, the neighborhood of E = 4 consists of a doubly infinite sequenceof intervals where the behavior alternates between stable and unstable (see Fig. 3.4). Thisis similar to the infinite sequence of transition energies for certain periodic orbits of a classof Hamiltonians studied in [21] and to the singly infinite sequence of transitions in the 2danharmonic oscillator as the coupling α goes from zero to infinity [82]: H anharm = 12 (cid:0) p + p (cid:1) + 14 (cid:0) q + q (cid:1) + α q q . (3.40)This accumulation of stable-to-unstable transition energies at the threshold for librational‘bound’ trajectories is also reminiscent of the quantum mechanical energy spectrum of Efimovtrimers that form a geometric sequence accumulating at the zero-energy threshold correspond-ing to arbitrarily weak two-body bound states with diverging S-wave scattering length [27]. ( E < ≤ E ≤ E (cid:96) , φ = arg λ monotonically increases from 0 to 2 π withgrowing energy and λ = e − iφ goes round the unit circle once clockwise. There is a stable tounstable phase transition at E (cid:96) . In the unstable phase E (cid:96) < E < E (cid:96) , σ > CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM Figure 3.4: Numerically obtained stability index of pendulum solutions showing approach to peri-odic oscillations between stable and unstable phases as E → ± . Equations (3.42) and (3.45) areseen to fit the data as E → ± . to real positive λ increasing from 1 to 1 . E (cid:96) . This pattern repeats so that the librationalregime 0 < E < E (cid:96) n +1 and E (cid:96) n denote the energies of the stable to unstable and unstable to stable transitions for n = 1 , , , . . . , then the widths w lun and w lsn +1 of the n th unstable and n + 1 st stable phasesare w lun = E (cid:96) n − E (cid:96) n − ≈ ( E (cid:96) − E (cid:96) ) × e − Λ ( n − and w lsn +1 = E (cid:96) n +1 − E (cid:96) n ≈ ( E (cid:96) − E (cid:96) ) × e − Λ ( n − . (3.41)Here, E (cid:96) − E (cid:96) ≈ e − . (1 − e − . ) and E (cid:96) − E (cid:96) ≈ e − . (1 − e − . ) are the lengths of the firstunstable and second stable intervals while Λ ≈ . 11 + 4 . 34 = 5 . 45 is the combined period ona log scale. The first stable phase has a width E (cid:96) − ≈ − e − . that does not scale likethe rest. Our numerically obtained stability index (see Fig. 3.4) is well approximated by σ ≈ . 22 cos (cid:20) √ − E ) + . (cid:21) + . 22 as E → − . (3.42)On the other hand, σ ( E ) ∼ − O ( E ) when E → ( E > E ≥ E r , the rotational pendulum solutions are stable. In fact,as E decreases from ∞ to E r , λ = e − iφ goes counterclockwise around the unit circle from1 to − 1. There is a stable to unstable transition at E r . As E decreases from E r to E r , λ is real and negative, decreasing from − − . − E r ≥ E ≥ E r where λ completes its passagecounterclockwise around the unit circle reaching 1 at E r . The last phase of this first cycle .3. REDUCTIONS TO ONE DEGREE OF FREEDOM E r and E r where λ is real and positive, increasingfrom 1 to 1 . λ made complete revolutions around the unitcircle in each stable phase and was always positive in unstable phases. This is reflected inthe stability index overshooting both 2 and − E = 4, given bystable energies = [ E r , ∞ ) ∞ (cid:91) n =1 (cid:2) E r n +1 , E r n (cid:3) and unstable energies = ∞ (cid:91) n =1 (cid:0) E r n , E r n − (cid:1) . (3.43)As before, with the exception of the two stable and one unstable intervals of highest energy,the widths of the stable and unstable energy intervals are approximately constant on a logscale: w run = E r n − − E r n ≈ ( E r − E r ) × e − Λ ( n − and w rsn +1 = E r n − E r n +1 ≈ ( E r − E r ) × e − Λ ( n − (3.44)for n = 2 , , · · · . Here, E r − E r ≈ e − . (1 − e − . ) and E r − E r ≈ e − . (1 − e − . ) arethe lengths of the second unstable and third stable intervals while Λ ≈ . . . E ≥ E r ≈ . E r > E > E r ≈ . 48 has width1 . > . E r ≥ E ≥ E r ≈ . 01 has a less than typical width 3 . < . σ ≈ − . 11 cos (cid:20) √ E − − . (cid:21) as E → + (3.45)while σ ( E ) ∼ − O (1 /E ) when E → ∞ . Since the pendulum solutions form a one parameter family of periodic orbits (0 , ϕ , p , p )with continuously varying time periods, a perturbation tangent to this family takes a pen-dulum trajectory to a neighboring pendulum trajectory and is therefore neutrally stable.These perturbations span the 1-eigenspace span( v , v ) of the monodromy matrix, where v = (0 , , , 0) = ∂ ϕ and v = (0 , , , 2) = ∂ p + 2 ∂ p . The other two eigenvectors of M have a simple dependence on energy and thus help in ordering the eigenvalues λ and λ away from transitions. In the ‘unstable’ energy intervals( E (cid:96) , E (cid:96) ) ∪ ( E r , E r ) ∪ ( E (cid:96) , E (cid:96) ) ∪ ( E r , E r ) ∪ . . . , (3.46) M = diag(1 , , λ , /λ ) in the basis ( v , v , v + , v − ) where v ± = (2 a ( E ) , − a ( E ) , ± b ( E ) , M = diag(1 , , R φ ) in the complementary ‘stable’ energy intervals (0 , E (cid:96) ) ∪ ( E r , ∞ ) ∪ · · · . Here, R φ is the 2 × φ, sin φ | − sin φ, cos φ ). At the6 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM 12 4.24.2 4.2 4.54.5 668810 101010 4.5 - - - - - - φ φ Phase portrait of breathers G DRR LGLD TT Figure 3.5: Level contours of E on a phase portrait of the LG, LD and R families of isoscelesperiodic solutions. transition energies, either a or b vanishes so that v + and v − become collinear and continuityof eigenvectors with E cannot be used to unambiguously order the corresponding eigenvaluesacross transitions. For instance, the eigenvalue that went counterclockwise around the unitcircle for E < E (cid:96) could be chosen to continue as the real eigenvalue of magnitude eithergreater or lesser than one when E exceeds E (cid:96) . Pitfall in trigonometric and quadratic approximation at low energies: Interestingly,if for low energies (0 ≤ E (cid:28) g ), we use the simple harmonic/trigonometric approximation to(3.34), ¯ ϕ ≈ (cid:112) E/g sin ω t with ω = (cid:112) g/mr and E ≈ ( mr / 3) ˙¯ ϕ + g ¯ ϕ and approximatecos ¯ ϕ by 1 − ¯ ϕ / e ± iθ and e ± iφ where θ and φ monotonically increase from zero with energy up tomoderate energies. By contrast, as we saw above, two of the eigenvalues λ , are alwaysequal to one, a fact which is not captured by this approximation. We seek solutions where two of the separations remain equal at all times: θ i − θ j = θ j − θ k where ( i, j, k ) is any permutation of (1,2,3). Loosely, these are ‘breathers’ where one rotoris always at rest midway between the other two (see Fig. 3.3b). For definiteness, suppose θ − θ = θ − θ or equivalently ϕ = ϕ . Putting this in Eq. (3.9), we get a single evolutionequation for ϕ = ϕ = ϕ , mr ¨ ϕ = − g (sin ϕ + sin 2 ϕ ) , (3.47)which may be interpreted as a simple pendulum with an additional periodic force. As before,each periodic solution of this equation may, upon inclusion of CM motion, be used to obtainquasi-periodic solutions of the three-rotor problem.At E = 0, the isosceles solutions reduce to the ground state G. More generally, there are .3. REDUCTIONS TO ONE DEGREE OF FREEDOM E denoting energy in units of g , they are LG(oscillations around G ( ϕ = 0) for 0 ≤ E ≤ / 2) and LD (oscillations around D ( ϕ = π )for 4 ≤ E ≤ / 2) with monotonically growing time period which diverges at the separatrixat E = 9 / E > / 2, we have rotational modes R with time perioddiminishing with energy ( τ rot ( E ) ∼ π/ √ E as E → ∞ ). At very high energies, one rotor isat rest while the other two rotate rapidly in opposite directions. Eq. (3.47) may be reducedto quadrature by use of the conserved relative energy (3.12): E = mr ˙ ϕ + g (3 − ϕ − cos 2 ϕ ) . (3.48)For instance, in the case of the LG family, ω t √ √ (cid:90) u du (cid:112) u (2 − u )( u − u + E/ 2) (3.49)where u = 1 − cos ϕ . The relative angle ϕ may be expressed in terms of Jacobi ellipticfunctions. Putting (cid:15) = √ − E , ϕ ( t ) = arccos (cid:18) − Eη (cid:15) + (3 − (cid:15) ) η (cid:19) where η ( t ) = sn (cid:32) √ (cid:15)ω t √ , (cid:114) ( (cid:15) − − (cid:15) )8 (cid:15) (cid:33) . (3.50)It turns out that the periods of both LG and LD families are given by a common expression, τ lib ( E ) = 4 √ ω √ (cid:15) K (cid:32)(cid:114) − − E (cid:15) (cid:33) for 0 ≤ E ≤ . . (3.51)As E → . τ lib diverges as 2 (cid:112) / . − E ). The time period of rotational solutions(for E ≥ . 5) is τ rot ( E ) = 4 √ ω ( E − E ) − / K (cid:32)(cid:115) 12 + 6 − E √ E − E (cid:33) . (3.52) The stability of isosceles solutions as encoded in the stability index ( σ = tr M − 2) isqualitatively different from that of the pendulum solutions. In particular, there is only oneunstable to stable transition occurring at E ≈ . 97 (see Fig. 3.6). Indeed, by computing themonodromies, we find that both families LG and LD of librational solutions are unstable.The stability index σ LG grows monotonically from 2 to ∞ as the energy increases from0 to 4 . 5. In particular, even though arbitrarily low energy breathers are small oscillationsaround the stable ground state G, they are themselves unstable to small perturbations. Bycontrast, we recall that low energy pendulum solutions around G are stable. On the other8 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM Figure 3.6: Absolute value of the stability index of the isosceles breathers as a function of energy. hand, the LD family of breathers are much more unstable, indeed, we find that σ LD increasesfrom ≈ . × to ∞ for 4 < E < . 5. This is perhaps not unexpected, given that theyare oscillations around the unstable static solution D. The rotational breathers are unstablefor 4 . < E < . 97 with σ R growing from −∞ to − 2. These divergences of σ indicatethat isosceles solutions around E = 4 . E = 8 . 97, the rotational breathers are stable with σ R growing from − E → ∞ . This stability of the breathers is also evident from the Poincar´e sectionsof § ϕ = 0’at hyperbolic to elliptic fixed points as the energy is increased beyond E ≈ . 97 (see Fig.3.9-3.11). We now consider a geometric reformulation of the classical three-rotor problem, that suggeststhe emergence of widespread instabilities for E > E → ∞ . This indicates the presence of an ‘order-chaos-order’ transition which will be confirmed in § § L = (1 / m ij ˙ q i ˙ q j −V ( q ) may be regarded as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric g JM ij = ( E − V ) m ij which is conformal to the mass/kinetic metric m ij ( q ). The sectionalcurvatures of this metric have information on the behavior of nearby trajectories with posi-tive/negative curvature associated to (linear) stability/instability. For the three-rotor prob-lem, the JM metric on the ϕ - ϕ configuration torus is given by ds = 2 mr E − V )( dϕ + dϕ dϕ + dϕ ) , (3.53)where V = g [3 − cos ϕ − cos ϕ − cos( ϕ + ϕ )]. Letting f denote the conformal factor E − V and using the gradient and Laplacian defined with respect to the flat kinetic metric, .4. JACOBI–MAUPERTUIS METRIC AND CURVATURE Scalar curvature R of the JM metric on the Hill region of the ϕ - ϕ torus. In theregions shaded grey, | R | is very large. We see that R > E ≤ g but has both signs for E > g indicating instabilities. the corresponding scalar curvature (2 × the Gaussian curvature) is R = |∇ f | − f ∆ ff = g mr ( E − V ) × (cid:20) (cid:18) Eg − (cid:19) (cid:18) − V g (cid:19) + cos( ϕ − ϕ ) + cos(2 ϕ + ϕ ) + cos( ϕ + 2 ϕ ) (cid:21) . (3.54) For 0 ≤ E ≤ g , R is strictly positive in the classically allowed Hill region ( V < E ) anddiverges on the Hill boundary V = E where the conformal factor vanishes (see AppendixB.2 for a proof and the first two ‘bath-tub’ plots of R in Fig. 3.7). Thus the geodesic flowshould be stable for these energies. Remarkably, we also find a near absence of chaos in allPoincar´e sections for E (cid:46) . g (see Fig. 3.9 and 3.12a). We will see that Poincar´e surfacesshow significant chaotic regions for E > g . This is perhaps related to the instabilitiesassociated with R acquiring both signs above this energy. Indeed, for 4 g < E ≤ g/ 2, theabove ‘bath-tub’ develops sinks around the saddles D (0 , π ), D ( π, 0) and D ( π, π ) where R becomes negative, though it continues to diverge on the Hill boundary which is a union of twoclosed curves encircling the local maxima T( ± π/ , ± π/ E > g/ 2, the Hill regionexpands to cover the whole torus. Here, though bounded, R takes either sign while ensuringthat the total curvature (cid:82) T R (cid:112) det g ij dϕ dϕ vanishes. For asymptotically high energies,the JM metric tends to the flat metric E m ij and R ∼ /E → CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM return to regularity. The static solutions G, D and T lie on the boundary of the Hill regions corresponding to theenergies E G , D , T = 0, 4 g and 4 . g . We define the curvatures at G, D and T by letting E approach the appropriate limiting values in the following formulae: R (0 , = 6 gmr E , R (0 ,π ) , ( π, , ( π,π ) = − g/mr ( E − g ) and R ( ± π , ± π ) = − g/mr (2 E − g ) . (3.55)Thus R G = ∞ while R D = R T = −∞ indicating that G is stable while D and T are unstable.These results on geodesic stability are similar to those obtained from (3.24). Note that wedo not define the curvatures at G, D and T by approaching these points from within the Hillregions as these limits are not defined for G and T and gives + ∞ for D. On the other hand,it is physically forbidden to approach the Hill boundary from the outside. Thus we approachG, D and T by varying the energy while holding the location on the torus fixed. To study the transitions from integrability to chaos in the three-rotor problem, we use themethod of Poincar´e sections. Phase trajectories are constrained to lie on energy level setswhich are compact 3d submanifolds of the 4d phase space parametrized by ϕ , ϕ , p and p (cotangent bundle of the 2-torus). By the Poincar´e surface ‘ ϕ = 0’ at energy E (in unitsof g ), we mean the 2d surface ϕ = 0 contained in a level-manifold of energy. It may beparametrized by ϕ and p with the two possible values of p ( ϕ , p ; E ) determined by theconservation of energy. Similarly, we may consider other Poincar´e surfaces such as the onesdefined by ϕ = 0, p = 0, p = 0 etc. We record the points on the Poincar´e surface wherea trajectory that begins on it returns to it under the Poincar´e return map, thus obtaining aPoincar´e section for the given initial condition (IC). For transversal intersections, a periodictrajectory leads to a Poincar´e section consisting of finitely many points while quasi-periodictrajectories produce sections supported on a finite union of 1d curves. We call these twotypes of sections ‘regular’. By a chaotic section, we mean one that is not supported onsuch curves but explores a 2d region. In practice, deciding whether a numerically obtainedPoincar´e section is regular or chaotic can be a bit ambiguous in borderline cases when it issupported on a thickened curve (see Fig. 3.11e and around I in Fig. 3.9). We define thechaotic region of a Poincar´e surface at energy E to be the union of all chaotic sections atthat energy. .5. POINCAR ´E SECTIONS: PERIODIC ORBITS AND CHAOS ����������������� � �� ��� ���� �� � �� � � �������� � | ϕ � | ���������� �� ��������� ������� (a) (b) (c)(d) (e) Figure 3.8: (a) The trajectories (e.g., | ϕ | ) obtained via different numerical schemes cease to agreeafter t ∼ for the IC ϕ = 6 . ϕ = 3 . p = − . 90 and p = 1 . 87 with E = 9 . 98. (b, c, d,e) However, Poincar´e sections (with ≈ × points) obtained via different schemes are seen toexplore qualitatively similar regions when evolved till t = 10 (though not for shorter times ∼ ). To obtain Poincar´e sections, we implement the following numerical schemes: ODE45: explicitRunge-Kutta with difference order 5; RK4 and RK10: implicit Runge-Kutta with differenceorders 4 and 10 and SPRK2: symplectic partitioned Runge-Kutta with difference order 2.Due to the accumulation of errors, different numerical schemes (for the same ICs) sometimesproduce trajectories that cease to agree after some time, thus reflecting the sensitivity toinitial conditions. Despite this difference in trajectories, we find that the correspondingPoincar´e sections from all schemes are roughly the same when evolved for sufficiently longtimes (see Fig. 3.8). Moreover, we find a strong correlation between the degree to whichdifferent schemes produce the same trajectory and the degree of chaos as manifested inPoincar´e sections. As the agreement in trajectories between different schemes improves, thePoincar´e sections go from being spread over 2d regions to being concentrated on a finiteunion of 1d curves. Since ODE45 is computationally faster than the other schemes, theresults presented below are obtained using it. Furthermore, we find that for all ICs studied,all four Poincar´e sections on surfaces defined by ϕ = 0, ϕ = 0, p = 0 and p = 0 arequalitatively similar with regard to the degree of regularity or chaos. Thus, in the sequel, werestrict to the Poincar´e surface defined by ϕ = 0. We find that for E (cid:46) 4, all Poincar´e sections (on the surface ‘ ϕ = 0’) are nearly regular anddisplay left-right ( ϕ → − ϕ ) and up-down ( p → − p ) symmetries (see Fig. 3.9). Thoughthere are indications of chaos even at these energies along the periphery of the four stable2 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM Figure 3.9: Several Poincar´e sections in the energetically allowed ‘Hill’ region on the ‘ ϕ = 0’surface for E = 2 and 3. All sections (indicated by distinct colors online) are largely regular andpossess up-down and left-right symmetries. The Hill boundary is the librational pendulum solution ϕ = 0. P, I and C indicate pendulum, isosceles and choreography periodic solutions. More carefulexamination of the vicinity of the I s shows small chaotic sections. lobes (e.g., near the unstable isosceles fixed points I ), chaotic sections occupy a negligibleportion of the Hill region. Chaotic sections make their first significant appearance at E ≈ ϕ → − ϕ symmetry (though not p → − p )seems to be restored when E (cid:38) . 4. The lack of p → − p symmetry at high energies is notunexpected: rotors at high energies either rotate clockwise or counter-clockwise.At moderate energies E (cid:38) 4, we observe that all chaotic sections (irrespective of the ICs)occupy essentially the same region, as typified by the examples in Fig. 3.11. At somewhathigher energies (e.g. E = 14), we find chaotic sections that fill up both the entire chaoticregion and portions thereof when trajectories are evolved up to t = 10 . At yet higherenergies (e.g. E = 18, Fig. 3.11e), there is no single chaotic section that occupies the entirechaotic region as the p → − p symmetry is broken. For a range of energies beyond 4 , we find that the area of the chaotic region increases with E (see Fig. 3.10 and 3.11). At E ≈ . 5, the chaotic region coincides with the energeticallyallowed portion of the Poincar´e surface (see Fig. 3.11c). Beyond this energy, chaotic sectionsare supported on increasingly narrow bands (see Fig. 3.11e). This progression towards regularsections is expected since the system acquires an additional conserved quantity in the limit E → ∞ . To quantify these observations, we find the ‘fraction of chaos’ f by exploiting thefeature that the density of points in chaotic sections is roughly uniform for all energies onthe ‘ ϕ = 0’ surface (this is not true for most other Poincar´e surfaces). Thus f is estimatedby calculating the fraction of the area of the Hill region covered by chaotic sections (see .5. POINCAR ´E SECTIONS: PERIODIC ORBITS AND CHAOS (a) (b) (c) Figure 3.10: Several Poincar´e sections on the ‘ ϕ = 0’ surface in the vicinity of E = 4 wherethe chaotic region (shaded, yellow online) makes its first significant appearance. Distinct sectionshave different colors online. On each surface, one sees breaking of both up-down and left-rightsymmetries. Aside from a couple of exceptions on the E = 4 surface, the set of ICs is left-rightand up-down symmetric. The boundary of the Hill region on the ‘ ϕ = 0’ Poincar´e surface is the ϕ = 0 pendulum solution. It becomes disconnected for E > Appendix B.3 and Fig. 3.12a).The near absence of chaos is reflected in f approximately vanishing for E (cid:46) . 8. Thereis a rather sharp transition to chaos around E ≈ f ≈ E = 3 . 85, 4and 4 . 1; see lower inset of Fig. 3.12a). This is a bit unexpected from the viewpoint of KAMtheory and might encode a novel mechanism by which KAM tori break down in this system.Thereafter, f rapidly rises and reaches the maximal value f ≈ E ≈ . 33. As illustratedin the upper inset of Fig. 3.12a, this ‘fully chaotic’ phase persists up to E ≈ . 6. Interestingly,we find that for this range of energies, f ≈ § § E r ≈ . E ≈ . f decreases gradually to zero as E → ∞ . Interestingly, the sharp transition to chaos at E ≈ § E < E > 4. It is noteworthy that the stable to unstable transitionenergies in pendula also accumulate from both sides at E = 4 (see Fig. 3.4).4 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM (a) (b) (c)(d) (e) Figure 3.11: The up-down symmetry remains broken, though the left-right symmetry is restoredon Poincar´e plots at higher energies. The periodic orbits corresponding to points marked C arechoreographies for E (cid:46) . ϕ = 0 ’ Poincar´e surface Here, we identify the points on the Poincar´e surface corresponding to the periodic pendulumand isosceles solutions. Remarkably, careful examination of the Poincar´e sections also leads usto a new family of periodic ‘choreography’ solutions which are defined and discussed furtherin § The ϕ = 0 pendulum solutions are everywhere tangent to the Poincar´e surface ‘ ϕ = 0’ andinterestingly constitute the ‘Hill’ energy boundary (see Fig. 3.9-3.11). [Nb. This connectionbetween pendulum solutions and the Hill boundary is special to the surfaces ‘ ϕ = 0’ and‘ ϕ = 0’.] By contrast, the other two classes of pendulum trajectories ( ϕ = 0 and ϕ + ϕ =0) are transversal to this surface, meeting it at the pendulum points P(0 , ± (cid:112) E/ 3) halfway .5. POINCAR ´E SECTIONS: PERIODIC ORBITS AND CHAOS (a) (b) Figure 3.12: (a) Energy dependence of the area of the chaotic region on the ‘ ϕ = 0’ Poincar´esurface as a fraction of the area of the Hill region. (b) Various Poincar´e surfaces showing globalchaos at E = 5 . to the boundary from the origin. These are period-2 and period-1 fixed points for librationaland rotational solutions respectively. Examination of the Poincar´e sections indicates thatpendulum solutions must be stable for E (cid:46) . E (cid:38) . § E → ± . Additionally, by considering initial conditionsnear the pendulum points, we find that the pendulum solutions lie within the large chaoticsection only between E ≈ . E ≈ . Unlike pendula, all isosceles periodic orbits intersect the ‘ ϕ = 0’ surface transversally atpoints on the vertical axis. Indeed, the breathers defined by ϕ = ϕ and ϕ + 2 ϕ = 0intersect the surface at the isosceles points I (0 , ±√ E ) which form a pair of period-2 fixedpoints for E < . ϕ + 2 ϕ = 0 intersect the surface at the period-1 fixed point at theorigin. In agreement with the conclusions of § . (cid:46) E (cid:46) . The period-2 fixed points C at the centers of the right and left lobes on the Poincar´e surfacesof Fig. 3.9 and 3.10 correspond to a new family of periodic solutions. Evidently, they gofrom being stable to unstable as the energy crosses E ≈ . 33. We argue in § CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM ϕ ( t ) ϕ ( t ) - - ϕ tChoreography at E = (a) E / g τ Time period of Choreographies (b) Figure 3.13: (a) A non-rotating choreography at E = 4 g showing that the time lag between ϕ and ϕ is one-third the period. (b) The time period 3 τ of non-rotating choreographies as a functionof energy indicating divergence at E ≈ . g . are choreographies for E (cid:46) . Choreographies are an interesting class of periodic solutions of the n -body problem whereall particles follow the same closed curve equally separated in time [57]. The Lagrangeequilateral solution where three equal masses move on a common circle and the stable zero-angular momentum figure-8 solution discovered by C. Moore [59] (see also [14]) are perhapsthe simplest examples of choreographies in the equal mass gravitational three-body problem.Here, we consider choreographies in the three-rotor problem where the angles θ i ( t ) of thethree rotors may be expressed in terms of a single 3 τ -periodic function, say θ ( t ): θ ( t ) = θ ( t + τ ) and θ ( t ) = θ ( t + 2 τ ) . (3.56)This implies that the CM and relative coordinates ϕ , ϕ ( t ) and ϕ ( t ) = ϕ ( t + τ ) must be3 τ periodic (see Fig. 3.13a) and satisfy the delay algebraic equation ϕ ( t ) + ϕ ( t + τ ) + ϕ ( t + 2 τ ) = θ − θ + θ − θ + θ − θ ≡ π. (3.57)The EOM (3.9) become 3 mr ¨ ϕ = 0 and the pair of delay differential equations mr ¨ ϕ ( t ) = − g (cid:2) ϕ ( t ) − sin ϕ ( t + τ ) + sin( ϕ ( t ) + ϕ ( t + τ )) (cid:3) and mr ¨ ϕ ( t ) = mr ¨ ϕ ( t + τ )= − g (cid:2) ϕ ( t + τ ) − sin ϕ ( t ) + sin( ϕ ( t ) + ϕ ( t + τ )) (cid:3) . (3.58)In fact, the second equation in (3.58) follows from the first by use of the delay algebraicequation (3.57). Moreover, using the definition of ϕ , the constant angular velocity of theCM ˙ ϕ = 1 τ [ ϕ ( t + τ ) − ϕ ( t )] = − τ [ ϕ ( t ) + ϕ ( t + τ ) + ϕ ( t + 2 τ )] . (3.59) .6. CHOREOGRAPHIES τ periodic triple ϕ , , satisfying (3.57), (3.58) and (3.59) leads toa choreography of the three-rotor system. Thus, to discover a choreography we only needto find a 3 τ -periodic function ϕ satisfying (3.57) and the first of the delay differentialequations (3.58) with the period 3 τ self-consistently determined. Now, it is easy to showthat choreographies cannot exist at asymptotically high (relative) energies. In fact, at highenergies, we may ignore the interaction terms ( ∝ g ) in (3.58) to get ϕ ( t ) ≈ ωt + ϕ (0) for | ω | (cid:29) 1. However, this is inconsistent with (3.57) which requires 3 ωt ≡ π at alltimes. On the other hand, as discussed below, we do find examples of choreographies at lowand moderate relative energies. Uniformly rotating (at angular speed Ω ) versions of the static solutions G and T (but not D)(see § θ ( t ) = Ω t and τ = 2 π/ Ω for G and τ = 2 π/ 3Ω for T where Ω is arbitrary. In the case of G, though allparticles coincide, they may also be regarded as separated by τ . The energies (3.12) of thesetwo families of choreographies come from the uniform CM motion and a constant relativeenergy: E (G)tot = 32 mr Ω and E (T)tot = 32 mr Ω + 9 g . (3.60)These two families of choreographies have the scaling property: if θ ( t ) with period 3 τ de-scribes a choreography in the sense of (3.56), then θ ( at ) with period | τ /a | also describesa choreography for any real a . It turns out that the above two are the only such ‘scaling’families of choreographies. To see this, we note that both θ ( t ) and θ ( at ) must satisfy thedelay differential equation¨ θ ( t + τ ) − ¨ θ ( t ) = − gmr [2 sin( θ ( t + τ ) − θ ( t )) − sin( θ ( t ) − θ ( t − τ )) + sin( θ ( t + τ ) − θ ( t − τ ))](3.61)implying that either a = 1 or ¨ θ ( t + τ ) = ¨ θ ( t ). However, the latter implies that ˙ θ ( t + τ ) − ˙ θ ( t ) = − ˙ ϕ ( t ) is a constant which must vanish for the delay algebraic equation (3.57) to besatisfied. Consequently, ˙ ϕ must also vanish implying that the choreography is a uniformlyrotating version of G or T. Remarkably, we have found another 1-parameter family of choreographies (e.g., Fig. 3.13a)that start out as small oscillations around G. At low energies, they have a period 3 τ = 2 π/ω and reduce to ϕ ( t ) ≈ (cid:115) E g sin( ω ( t − t )) for E (cid:28) g (3.62)where ω = (cid:112) g/mr . It is easily verified that (3.57) is identically satisfied while (3.58)is satisfied for E (cid:28) g . Moreover, using (3.59), we find that the angular speed ˙ ϕ of the8 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM CM must vanish for (3.62) so that the energy is purely from the relative motion. The phasetrajectory corresponding to (3.62) intersects the ϕ = 0 Poincar´e surface at the pair ofperiod-2 fixed points C( ± (cid:112) E/ g, 0) which lie at the centers of the left and right stable‘lobes’ pictured in Fig. 3.9 at E = 2 g and 3 g .More generally, we numerically find that when the ICs are chosen at the stable fixedpoints at the centers of these lobes, the trajectories are a one-parameter family of choreogra-phies ϕ ( t ; E ) varying continuously with E up to E ≈ . 33. It can be argued that thesechoreographies are non-rotating (involve no CM motion). Indeed, from (3.59) and (3.57),we must have 3 τ ˙ ϕ ≡ π , implying that ˙ ϕ cannot jump discontinuously. Since,3 τ ˙ ϕ = 0 as E → E is continuously increased from0 to 5 . 33. Though we do not study their stability here by the monodromy approach, thePoincar´e sections (see Fig. 3.9 and 3.10) indicate that they are stable. As shown in Fig.3.13b, the time period 3 τ grows monotonically with E and appears to diverge at E ≈ . § E (cid:38) . ϕ = 0 ’ Poincar´e surface become unstable andlie in a chaotic region (see Fig. 3.11), preventing us from finding such a choreography, if itexists, using the above numerical technique. As argued before, choreographies are forbiddenat very high energies. For instance, on the ‘ ϕ = 0’ Poincar´e surface at E = 18 (see Fig.3.11e), the analogues of the C points correspond to unstable periodic orbits which are not choreographies. In fact, we conjecture that this family of periodic solutions ceases to be achoreography beyond E ≈ . In § . g ≤ E ≤ . g ) and conjectured ergodicbehavior. Intriguingly, the beginning of this band coincides with the divergence in the periodof the non-rotating choreographies which additionally cease to exist above this energy (seeFig. 3.14). Similarly, the cessation of this band coincides with the energy at which pendulabecome stable. In this section, we provide evidence for ergodicity in this band by comparingdistributions of ϕ , and p , on constant energy hypersurfaces (weighted by the Liouvillemeasure) with their distributions along generic (chaotic) numerically determined trajectories.For ergodicity, the distribution along a generic trajectory (over sufficiently long times) shouldbe independent of initial condition and tend to the corresponding distribution over the energyhypersurface [3,31]. We also examine the rate of approach to ergodicity in time and deviationsfrom ergodicity outside the band of global chaos. Our numerical and analytical results, whileindicative of ergodic behavior, are nonetheless not sufficient to establish it, since we examineonly a restricted set of observables. .7. ERGODICITY IN THE BAND OF GLOBAL CHAOS Approach to the band of global chaos (5 . g ≤ E ≤ . g ) on the Poincar´e surface ϕ = 0 . The last elliptic islands to cease to exist (as E → . g − ) are around choreographies (C)and the first elliptic islands to open up (when E exceeds 5 . g ) are around pendula (P) which alsooccur along the Hill boundary. Isosceles solutions intersect this surface at the points marked I . Distribution along generic trajectories: By the distribution function of a dynamicalvariable F ( p, ϕ ) (such as p or ϕ ) along a given trajectory parametrized by time t , wemean (cid:37) F ( f ) = lim T →∞ T (cid:90) T δ ( F ( p ( t ) , ϕ ( t )) − f ) dt. (3.63)The time average of F along the trajectory is then given by the first moment (cid:104) F (cid:105) t = (cid:82) f (cid:37) F ( f ) df . In practice, to find the distribution of F , we numerically evolve a trajectorystarting from a random initial condition (IC) and record the values f of F at equally spacedintervals of time (say, ∆ t = . 25) up to t max = 3 × in units where g = m = r = 1. For such t max and for energies in the globally chaotic band, we find that the histograms of recordedvalues approach asymptotic distributions (see Fig. 3.15) that are largely independent of thechoice of ∆ t and ICs. Distributions over energy hypersurfaces: The ensemble average (cid:104)·(cid:105) e of a dynamicalvariable F ( p, ϕ ) at energy E is defined with respect to the Liouville volume measure onphase space. Since ϕ i and p j are canonically conjugate, we have (cid:104) F (cid:105) e = 1 V E (cid:90) F δ ( H − E ) dϕ dϕ dp dp where V E = (cid:90) δ ( H − E ) dϕ dϕ dp dp (3.64)is the volume of the H = E energy hypersurface M E . More generally, the distribution of F ( p, ϕ ) over the energy E hypersurface weighted by the Liouville measure is the followingphase space integral: ρ F,E ( f ) = 1 V E (cid:90) δ ( F ( p, ϕ ) − f ) δ ( H − E ) dϕ dϕ dp dp . (3.65)Loosely, it is like the Maxwell distribution of speeds in a gas. We will often omit the subscripts F and/or E when the observable and/or the energy are clear from the context. By definition,0 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM the above distribution is a probability density: (cid:82) ρ ( f ) df = 1. The ensemble average (cid:104) F (cid:105) e isits first moment: (cid:104) F (cid:105) e = (cid:90) f ρ F,E ( f ) df. (3.66)To find distributions over an energy hypersurface M E , we need to integrate over it. Forinstance, to find the volume V E of the energy hypersurface, we observe that the Hamiltonian H = T + V is quadratic in p where T = p + p − p p mr and V ( ϕ , ϕ ) = g [3 − cos ϕ − cos ϕ − cos( ϕ + ϕ )] . (3.67)Hence, we cover M E by two coordinate patches parametrized by ϕ , ϕ and p with p ± = 12 (cid:18) p ± (cid:113) mr ( E − V ( ϕ , ϕ )) − p (cid:19) . (3.68)Using the factorization H − E = ( p − p +2 )( p − p − ), we evaluate the integral over p in Eq.(3.64) to arrive at V E = (cid:90) (cid:90) ( ϕ ,ϕ ) ∈H E dϕ dϕ p max (cid:90) − p max dp ( p +2 − p − ) (3.69)where p max = (cid:112) mr ( E − V ) / 3. Here, ϕ , are restricted to lie in the Hill region H E ( V ≤ E ). Interestingly, the integral over p is independent of ϕ and ϕ as well as E sothat p max (cid:90) − p max dp ( p +2 − p − ) = π √ ⇒ V E = π √ × Area( H E ) . (3.70)Here, Area( H E ) is the area of the Hill region with respect to the measure dϕ dϕ . It is amonotonically increasing function of E and saturates at the value 4 π for E ≥ . ϕ - ϕ torus. We now derive formulae for distributions overenergy hypersurfaces. Distribution of angles: The joint distribution function of ϕ and ϕ is given by ( p ± areas in (3.68)) ρ E ( ϕ , ϕ ) = 1 V E (cid:90) δ ( H − E ) dp dp = 1 V E p max (cid:90) − p max dp ( p +2 − p − ) = πV E √ , (3.71)since from (3.70), the integral over p is π/ √ E and ϕ . In other words, ( ϕ , ϕ )is uniformly distributed on the Hill region. Furthermore, for E ≥ . 5, the Hill region is thewhole torus and ρ E ( ϕ , ϕ ) = 1 / π . Thus, ϕ and ϕ are each uniformly distributed on[0 , π ] for E ≥ . 5. Fig. 3.15a shows that the distributions of ϕ and ϕ along a trajectorywith energy E = 5 . .7. ERGODICITY IN THE BAND OF GLOBAL CHAOS ���� ��� �������� �������� ϕ � ��� � � ������������� ( � ) � = ���� ρ ( ϕ � ) - π - π � = �� ρ ( ϕ � ) - π - π � = ���� ρ ( ϕ � ) - π - π � = �� ρ ( ϕ � ) - π - π � = ���� ρ ( ϕ � ) - π - π Figure 3.15: Distribution along generic trajectories (yellow, lighter) and distribution over constantenergy hypersurface (black, darker) of (a) relative angle ( ϕ ) and (b) relative momentum ( p ) fora range of increasing energies with m = r = g = 1. The horizontal axis is ϕ in (a) and p in (b).Note that ϕ and ϕ have the same distributions as do p and p . The distribution along a generic(chaotic) trajectory is found to be insensitive to the IC chosen. The momentum distribution overconstant energy hypersurfaces transitions from a Wigner semi-circle to a bimodal distribution withincreasing energy. The two distributions agree only in the band of global chaos (5 . ≤ E ≤ . p =- p =- p =- p =- p = p = p = p = p = Figure 3.16: The energetically allowed portion (shaded gray) of the ϕ - p Poincar´e surface for asequence of increasing values of p at E = 5 . m = r = g = 1 . Oneach plot, the horizontal axis is ϕ ∈ [ − π, π ] and the vertical axis is p ∈ [ − , ρ E ( p ) is the Liouville area of the shaded region. It is plausible that ρ E ( p )is even and that as p goes from 0 to p max = (cid:112) mr E/ ≈ . ρ E ( p ) initially increases froma non-zero local minimum, reaches a maximum and then drops to zero as shown in the E = 5 . Distribution of momenta: The momentum distribution functions turn out to be moreintricate. Due to the 1 ↔ ρ E ( p ) and ρ E ( p ) are equal and given by the marginal distribution ρ E ( p ) = 1 V E (cid:90) δ ( H − E ) dϕ dϕ dp = 1 V E (cid:90) (cid:90) ( ϕ ,ϕ ) ∈H E,p dϕ dϕ p +2 − p − . (3.72)Here, H E,p is the portion of the ϕ - ϕ torus allowed for the given values of E and p . Since p ± must be real, from (3.68) we see that 4 mr ( E − V ) − p ≥ V ≤ E − p / mr .Thus, ϕ and ϕ must lie in the Hill region for the modified energy E (cid:48) = E − p / mr .For this Hill region to be non-empty, we must have E (cid:48) ≥ 0. Thus, the distribution function ρ E ( p ) is supported on the interval [ − (cid:112) mr E/ , (cid:112) mr E/ 3] and is given by ρ E ( p ) = 1 V E (cid:90) (cid:90) H E (cid:48) dϕ dϕ (cid:112) mr ( E (cid:48) ( p ) − V ) . (3.73)2 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM On account of E (cid:48) ( p ) being even, ρ E ( p ) = ρ E ( − p ). Upon going to Jacobi coordinates ϕ ± = ( ϕ ± ϕ ) / 2, the integral over ϕ − can be expressed in terms of an incomplete ellipticintegral of the first kind. Though the resulting formulae are lengthy in general, for lowenergies ρ E ( p ) turns out to be the Wigner semi-circular distribution (see Fig. 3.15b). Indeed,upon going to Jacobi coordinates and using the quadratic approximation for the potential V low = 3 gϕ + gϕ − , we find that at low energies, the Hill region H E (cid:48) is the elliptical disk3 gϕ + gϕ − ≤ E (cid:48) ( p ). Thus, V E = π √ × Area( H E ) = 2 π E g for E (cid:28) g (3.74)leading to the Wigner semi-circular distribution ρ E ( p ) ≈ V E (cid:90) (cid:90) H E (cid:48) dϕ + dϕ − (cid:112) mr ( E (cid:48) ( p ) − V low ) = 32 πmr E (cid:114) mr E − p for E (cid:28) g. (3.75)For larger E , we perform the integral (3.73) numerically. Fig. 3.15b shows that the dis-tribution goes from being semi-circular to bimodal as E crosses 4 g . Loosely, ρ E ( p ) is theanalogue of the Maxwell distribution for the relative momenta of the three-rotor problem.Fig. 3.16 provides a qualitative explanation of the bimodal shape of ρ E ( p ) for an energyin the band of global chaos. Fig. 3.15b shows that the distribution of p along a generictrajectory closely matches its distribution ρ E ( p ) over the constant energy hypersurface inthe band of global chaos (5 . ≤ E ≤ . 6) but deviates at other energies, providing evidencefor ergodic behavior in this band. To examine the rate of approach to ergodicity for energies in the band of global chaos, wecompare ensemble averages of variables such as cos ϕ and p with their time averages overincreasingly long times. Ensemble average: The ensemble average (cid:104)·(cid:105) e of a variable F at energy E defined in(3.64) reduces to (cid:104) F (cid:105) e = 1 V E (cid:90) (cid:90) ( ϕ ,ϕ ) ∈H E dϕ dϕ p max (cid:90) − p max dp F ( ϕ , ϕ , p , p +2 ) + F ( ϕ , ϕ , p , p − )2( p +2 − p − ) (3.76)upon using the factorization H − E = ( p − p +2 )( p − p − ) to evaluate the integral over p .Since for E ≥ . ϕ and ϕ are independently uniformly distributed on [0 , π ], we have (cid:104) cos m ϕ cos n ϕ (cid:105) e = (cid:104) cos m ϕ (cid:105) e (cid:104) cos n ϕ (cid:105) e (3.77)with (cid:104) cos n ϕ (cid:105) e = (2 n )!2 n ( n !) and the odd moments vanishing. Remarkably, the phase spaceaverages of momentum observables are also exactly calculable for E ≥ . (cid:104) p (cid:105) e = 2 E/ − , (cid:104) p (cid:105) e = 2 E / − E + 7 , .7. ERGODICITY IN THE BAND OF GLOBAL CHAOS (a) (b) Figure 3.17: (a) Time averages (cid:104) p (cid:105) t and (cid:104) cos ϕ (cid:105) t as a function of time T for 35 randomlychosen trajectories at E = 5 . 5. They are seen to approach the corresponding ensemble averages( (cid:104)·(cid:105) e indicated by thick black lines) as time grows. (b) Root mean square deviation (over 35 chaoticinitial conditions) of time averages from the corresponding ensemble average as a function of time T for E = 5 . ϕ , cos ϕ , p and p . The fitsshow a T − / approach to ergodicity. (cid:104) p p (cid:105) e = E / − E + 7 / (cid:104) p (cid:105) e = 20 E / − E / E/ − / . (3.78)Though we restrict to E ≥ . . ≤ E ≤ . alone we can expect ergodic behavior.To compare with time averages, for each energy, we pick N traj = 35 random ICs (onthe ϕ = 0 surface) and evolve them forward. As Fig. 3.17a indicates, though the timeaverages ( T (cid:82) T F dt ) display significant fluctuations at early times, they have approachedtheir asymptotic values by T = 10 . To estimate the rate of approach to ergodicity, wecompute the root mean square deviation σ ( T ) of the time average from the ensemble averageas a function of time: σ ( T ) = 1 N traj (cid:88) a ( (cid:104) F (cid:105) t,a ( T ) − (cid:104) F (cid:105) e ) where (cid:104) F (cid:105) t,a ( T ) = 1 T (cid:90) T F ( t (cid:48) a ) dt (cid:48) a (3.79)is the time average over the a th trajectory. Fig. 3.17b shows that for several variables F = cos ϕ , p etc., the mean square deviation decays roughly as the reciprocal of time, σ ∼ / √ T , as expected of an ergodic system where correlations decay sufficiently fast asshown in Appendix B.4 (see also [24] for a stochastic formulation).Finally, we examine the approach to ergodicity as the energy approaches the band ofglobal chaos 5 . (cid:46) E (cid:46) . 6. To this end, we compare the ensemble averages of a fewvariables with their time averages for 35 randomly chosen chaotic trajectories over a rangeof energies. Fig. 3.18 shows that the time averages of cos ϕ and p agree reasonably well4 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM - - - Difference between time & ensemble averages 〈 cos2 ϕ 〉 t - 〈 cos2 ϕ 〉 e Band ofglobal chaos - - Difference between time & ensemble averages 〈 p 〉 t - 〈 p 〉 e Band ofglobal chaos Figure 3.18: Difference between time averages (cid:104)·(cid:105) t over a time T = 10 (for 35 randomly chosenchaotic trajectories) and ensemble average (cid:104)·(cid:105) e for cos ϕ and p indicating ergodicity in the bandof global chaos 5 . ≤ E ≤ . (cid:104)·(cid:105) t − (cid:104)·(cid:105) e at a fixed energy is due to the finiteness of T . However, this spread issmall compared to the average values (cid:104) cos ϕ (cid:105) e = . (cid:104) p (cid:105) e = 2 E / − E + 7 demonstratingthat time averages over distinct chaotic trajectories converge to a common value. Note that thespread in (cid:104) p (cid:105) t − (cid:104) p (cid:105) e increases with E as the average values themselves increase with E . with their ensemble averages in the band of global chaos. At lower and higher energies, thereare discernible deviations from the ensemble averages, showing ergodicity breaking. (a) For E slightly outside the band of global chaos, we find that there is a single chaotic region(see Fig. 3.14), and time averages along trajectories from this region converge to a commonvalue which however differs from the ensemble average over the whole energy hypersurface(see Fig. 3.18). (b) At energies significantly outside the band of global chaos, there can beseveral distinct chaotic regions (see Fig. 3.11e). We find that time averages of an observablealong chaotic trajectories from these distinct regions generally converge to different values,none of which typically agrees with the ensemble average over the whole energy hypersurface. In § φ t on the energy hypersurface M E of the phase space is said to be stronglymixing if for all subsets A, B ⊆ M E with positive measures ( µ ( A ) > µ ( B ) > t →∞ µ ( φ t ( B ) ∩ A ) = µ ( B ) × µ ( A ) /µ ( M E ) (3.80)where µ is the Liouville volume measure on M E [3, 31]. To numerically examine whetherthe dynamics of three-rotors is mixing in the band of global chaos, we work in units where m = r = g = 1 and consider a large number N (= 1 . × ) of random ICs with energy E in a small initial region of phase space (e.g., | ϕ , | , | p | < . 05 with p = p +2 (3.68) determinedby E ). The trajectories are numerically evolved forward in time and their locations recordedat discrete time intervals (e.g., t = 10 , 20, · · · , 300). If the dynamics is mixing, then in thelimit N → ∞ and t → ∞ , the number of trajectories located at time t in a Liouville volume .8. MIXING IN THE BAND OF GLOBAL CHAOS timesApproach to mixing with time at E = (a) �������� �� ������ ���� ������ E = = 300 E = = E = = E = = (b) Figure 3.19: Histograms of number of trajectories n i ( t ) in each cell i of an energy hypersurface.To facilitate comparison across energies and numbers of ICs considered, the histograms of ˜ n i ( t ) =( n i ( t ) V E ) / ( µ i N ) (see Eq. 3.82) are displayed. For the flow to be mixing, the histograms shouldstrongly peak around ˜ n i ( t ) = 1. Fig. (a) shows the approach to mixing in time at an energy E = 5 . t = 300)showing how the flow becomes mixing as we approach the band of global chaos (represented hereby E = 5 . V must equal N V /V E where V E is the Liouville volume of the energy hypersurface. Poincar´esections (see Fig. 3.14) as well as investigations of ergodicity in § . ≤ E ≤ . V E = 4 π / √ 3, a formula that holds for any E ≥ . V g .The Liouville volumes of these cells are not equal, so we denote by µ i the Liouville volumeof the i th cell. In practice, we take cells of linear dimensions 2 π/d each in ϕ and ϕ and2 p max1 /d in p where d = 40 is the number of subdivisions and p max1 the maximal value of p corresponding to energy E . Though we compute µ i exactly, it is approximately V g × theLiouville density at the center of the i th cell: µ i ≈ p +2 − p − ) × πd × πd × p max1 d (3.81)where p ± (3.68) are evaluated at the center of the cell. Cells that lie outside or straddle theboundary of the energy hypersurface are not considered. At various times, we record theinstantaneous locations of the trajectories and count the number n i ( t ) of trajectories that liein the cell i . If the dynamics is mixing, the number of trajectories in the i th cell should be n i = N × µ i V E . (3.82)To test the mixing hypothesis and the rate of approach to mixing, we plot in Fig. 3.19at various times t = 10 , , · · · , n i ( t ). To be more precise, we plot a6 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM histogram of ˜ n i ( t ) = n i ( t ) V E / ( µ i N ) so that the expected mean is 1, to facilitate comparisonacross energies, times and numbers of ICs considered. At very early times ( t (cid:46) t increases, we observe from Fig. 3.19a that the histograms shift, and becomeprogressively narrower, peaking around the expected value of 1 with the expected width (seeFig. 3.20). This provides evidence for mixing in the regime of global chaos. In Fig. 3.19b,we compare these histograms at sufficiently late times ( t = 300) for a range of energies andobserve significant departures from mixing for energies outside the band of global chaos. Infact, for energies such as E = 4 . E = 6, the histograms in Fig. 3.19b show threedistinct peaks corresponding to cells that are never visited and two other types of cells (inchaotic regions) that are visited with unequal frequencies (see Fig. 3.21). This characteristicdeparture from mixing with respect to the Liouville measure (even when restricted to chaoticregions) is also reflected in the two distinct densities of points in Poincar´e plots at suchenergies, as seen in Fig. 3.21. 50 100 150 200 250 300 - - time elapsed Approach to mixing:Drop in standard deviation of n ˜ i ( t ) l og ( s t de v ) Figure 3.20: Drop with time of the standard deviation of the distribution (see Fig. 3.19a) of thescaled number of trajectories ˜ n i ( t ) in each cell of the energy E = 5 . N cells ≈ × cells and N = 1 . × trajectories have been considered. The plotshows that the standard deviation has dropped to 0 . 066 at t = 300. This is close to the expectedstandard deviation 0 . 055 if the N trajectories were distributed uniformly among the N cells cells atthe instant considered. Here, we study the statistics of Poincar´e recurrence times to a three-dimensional cell in anenergy- E hypersurface of the phase space. For convenience, we choose the cell to be a cuboidof width w , e.g, − w/ ≤ ϕ , ϕ , p ≤ w/ p = p +2 (3.68) determined by energy fora cell centered at the origin. We choose a large number ( ∼ × ) of initial conditionsdistributed uniformly randomly within the cell and numerically evolve them forward in time.The recurrence time τ for a given trajectory is defined as the time from the first exit to the .9. RECURRENCE TIME STATISTICS Two distinct densities (shaded dark and light) of points (from trajectories for 0 ≤ t ≤ ) on chaotic sections of Poincar´e surfaces at E = 4 . next exit from the cell (see Fig. 3.22) [83]. Evidently, starting from the instant the trajectoryfirst exits the cell, τ is the sum of the times it spends outside the cell and while traversingthe cell. A histogram of the recurrence times (normalized to be a probability distribution) isthen plotted as in Fig. 3.23a. t n+1 t n 𝛕 = t n+1 - t n Figure 3.22: Recurrence time τ . For uniformly mixing dynamics, it is expected thatthis normalized distribution follows an exponential law(1 / ¯ τ ) e − τ/ ¯ τ where ¯ τ is the mean recurrence or relaxationtime [83]. As shown in Fig. 3.23, this exponential law forrecurrence times holds for energies in the band of globalchaos though there can be (sometimes significant) devia-tions for very small values of τ (e.g., τ (cid:46) (cid:28) ¯ τ ≈ w = .6 in Fig. 3.23d). These deviations could beattributed to a memory effect, the finite time that thesystem takes before the dynamics displays mixing (seeFig. 3.19a). Thus, ¯ τ is to be interpreted as the time constant in the above exponential lawthat best fits the distribution away from very small τ .A heuristic argument for the exponential law follows; for a more detailed treatment,see [4, 37] and references therein. We pick a large number N of ICs uniformly from a regionΩ of volume V Ω in an energy- E hypersurface of volume V E . They are evolved in time and8 CHAPTER 3. CLASSICAL THREE–ROTOR PROBLEM Recurrence time distributionat energy = τ Cell centered at originwith width = τ = c oun t (a) cell volume ( v ) τ slope = - / scaling law: τ ∝ v - / E = cell center at ϕ = .6, ϕ = p = .7 (b) 50 100 150 200 250 - - - - - Scaled recurrence time distributionat E = l og ( c oun t ) cell center at origin τ × v / τ * = (c) 50 100 150 200 250250 300 350 - - - - - Scaled recurrence time distributionat E = cell center at ϕ = .01, ϕ =- p = l og ( c oun t ) τ × v / τ * = (d) Figure 3.23: (a) Histogram of recurrence times (normalized to be a probability distribution) fora cubical cell centered at the origin ( p = ϕ = ϕ = 0) of the globally chaotic energy-5 . m = r = g = 1) hypersurface showing an exponential law (1 / ¯ τ ) exp( − τ / ¯ τ ) where ¯ τ is the fitted mean recurrence time. Note that ¯ τ ≈ 580 is much larger than the time scale of thelinearized system (1 /ω = (cid:112) mr / g ). (b) At any cell location, ¯ τ scales as the minus two-thirdspower of the Liouville volume of the cell, consistent with ergodicity. (c, d) Normalized histogramof (recurrence times) × (cell volume) / plotted on a log-linear scale for cells of various widths,showing a universal exponential distribution (1 /τ ∗ )exp( − τ /τ ∗ ) away from very small τ . The largerspread at large τ × v / is due to lower statistics. The rescaled fitted mean recurrence time τ ∗ varieswith cell location but only weakly depends on energy within the band of global chaos. their locations sampled at a temporal frequency ∆ . At each such instant, the probability ofreturning to Ω is p = V Ω /V E provided a sufficiently long time T has elapsed for correlationsto have died out. Suppose a fraction f of trajectories have not returned to Ω by this time T . Then, the probability that the first return time τ equals T + ∆ is P ( τ = T + ∆) = f p (leaving aside possible returns that the sampling at frequency ∆ does not detect). If ∆ ischosen large enough ( (cid:38) transit time across Ω), we also have P ( τ = T + 2∆) = f (1 − p ) p and similarly P ( τ = T + n ∆) = f (1 − p ) n − p for n = 1 , , · · · . In the limit N → ∞ , ∆ → V Ω → /p = ¯ τ fixed, and omitting prefactors (independent of t ) that go intothe normalization, P ( t ≤ τ ≤ t + dt ) ∝ lim ∆ → (1 − p ) t/ ∆ = e − t/ ¯ τ . (3.83) .9. RECURRENCE TIME STATISTICS Though ¯ τ varies with the width w , we find that when rescaled by the two-third power of theLiouville volume v of the cell, it becomes independent of cell size within the band of globalchaos . In other words, ¯ τ × v / = τ ∗ is constant for cells centered at a given location (see Fig.3.23b). Thus, as shown in Figs. 3.23c and 3.23d, the rescaled recurrence time distributionsfor various cell sizes, all follow the same exponential law for a given energy and cell center.This scaling law may be viewed as a 3d energy hypersurface analogue of the 2d phase spaceversion given in Eq. (36) of [62] as well as of the scaling law for the mean recurrence time ofthe second type in [29]. Heuristically, the mean recurrence time ¯ τ is inversely proportionalto the surface area ( ∼ v / ) of the cell and allows us to view the ‘attractor’ as being threedimensional, which is consistent with global chaos and ergodicity. On the other hand, we findthat the scaling exponent deviates from two-thirds in chaotic regions outside this band. Thisis to be expected since the dynamics at such energies is not mixing in such chaotic regions,as shown in Figs. 3.19b and 3.21.The above scaling law defines for us the scaled mean recurrence time τ ∗ for cells centeredat a given location of an energy hypersurface. We find that τ ∗ varies with location. Forinstance, for cells centered along an isosceles trajectory (see § τ ∗ display a reflection symmetry about the triple collision configuration and vary over therange 31 (cid:46) τ ∗ (cid:46) 56. On the other hand, within the band of global chaos, τ ∗ hardly varieswith energy for a given cell location. We also observe the absence of memory in the sense that the gaps between successive recur-rence times are uncorrelated. For instance, let us denote by τ and τ the first recurrencetime and the gap between second and first recurrence times for a given trajectory and cell,and define the the correlation coefficient r = [ (cid:104) τ τ (cid:105) − (cid:104) τ (cid:105)(cid:104) τ (cid:105) ] / ( σ σ ) . (3.84)The averages here are performed with respect to a random collection of trajectories and σ , denote the standard deviations of τ , . We find that | r | ≈ − − − (cid:28) . − . E = 5 . For ergodic systems defined by the iterations of a map, Kac’s Lemma implies that the mean first returntime to a cell is inversely proportional to the measure of the cell [38]. What we observe here is a continuoustime version of it. hapter 4Discussion In the first part of this thesis (Chapter 2), we investigate the planar three-body problemwith Newtonian and inverse-square potentials from a geometric viewpoint where trajectoriesare reparametrized geodesics of the Jacobi-Maupertuis metric on the configuration space.Symmetries are used to pass to quotients of the configuration space using the method ofRiemannian submersions. We study the near-collision dynamics and show that the geodesicformulation regularizes collisions in the inverse-square potential, though not for the Newto-nian potential. Explicit calculations are facilitated by a good choice of coordinates in whichKilling vector fields point along coordinate vector fields. By estimating scalar and sectionalcurvatures, we establish the presence of widespread geodesic instabilities. The results aresummarized in § § E = 4 g as in [21] and establish its asymptotic periodicity on a log scale, for instance by finding ananalytical expression for the stability index as Yoshida [82] does in the 2d anharmonic os-cillator of Eq. (3.40). This accumulation at the threshold for bound librational trajectorieswith diverging time periods and the periodicity on a log scale is reminiscent of the quantumenergy spectrum of Efimov trimers that accumulate via a geometric sequence at the two-body bound state threshold with diverging S-wave scattering length [27]. It would also beinteresting to explore a possible connection between this accumulation of transitions and theaccumulation of homoclinic points at a hyperbolic fixed point in a chaotic system. The natureof bifurcations [6] and local scaling properties [47] at these transitions are also of interest.In another direction, one would like to understand if there is any connection between theaccumulation of transition energies and the change in topology of the Hill region ( V ≤ E )of the configuration torus as E crosses the value 4 g at the three critical points (saddles D)of the Morse function V (see § E = 0 and ∞ ( g = ∞ , ϕ = 0’ Poincar´e surfacestrongly suggests that any integrable energy E I is either isolated or E I (cid:46) . g . However,even for low energies, we expect chaotic sections in the neighborhood of the isosceles points I (see Fig. 3.9). In fact, we conjecture that the three-rotor problem has no non-trivialintegrable energies unlike the 2d anharmonic oscillator [82].While we have provided a qualitative explanation for the shape of the momentum distri-bution over energy hypersurfaces in § ρ ( p ) as the energy is varied. In another di-rection, outside the band of global chaos, it would be interesting to determine whether thedynamics, when restricted to a chaotic region, is ergodic and/or mixing with respect to asuitable measure. In fact, Figs. 3.19b and 3.21 suggest that this measure cannot be theLiouville measure. In § τ ∗ to cells at a given location isfound to vary with the location on the energy hypersurface. It would be of interest to studythe nature of this variation and its physical implications. We also wonder whether globalchaos and ergodicity are to be found in the problems of four or more rotors.Unlike billiards and kicked rotors, the equations of the three-rotor system do not in-volve impulses/singularities. It would be interesting to identify other such continuous timeautonomous Hamiltonian systems that display global chaos and ergodicity. As noted, thethree-rotor problem may also be formulated as geodesic flow on a two-torus of non-constantJacobi-Maupertuis curvature. A challenging problem would be to try to extend the analytic2 CHAPTER 4. DISCUSSION treatments of ergodicity in geodesic flows on constant curvature Riemann surfaces to thethree-rotor problem.Finally, a deeper understanding of the physical mechanisms underlying the onset of chaosin this system would be desirable, along with an examination of quantum manifestations ofthe classical chaos and an exploration of ergodicity and recurrence in the quantum three-rotorsystem. ppendix AThree-body problem A.1 Some landmarks in the history of the three-bodyproblem We consider the problem of three point masses ( m a with position vectors r a for a = 1 , , m a d r a dt = (cid:88) b (cid:54) = a Gm a m b r b − r a | r b − r a | for a = 1 , , . (A.1)The three components of momentum P = (cid:80) a m a ˙ r a , three components of angular momentum L = (cid:80) a r a × p a and energy E = 12 (cid:88) a =1 m a ˙ r a − (cid:88) a
Euler’s and Lagrange’s periodic solutions of the three-body problem. The constantratios of separations are functions of the mass ratios alone. (a) Euler collinear solution where massestraverse Keplerian ellipses with one focus at the CM. (b) Euler’s solution where two equal masses m are in a circular orbit around a third mass M at their CM. (c) Lagrange’s periodic solution withthree bodies at vertices of equilateral triangles. (see Fig. A.1c). In the limiting case of zero angular momentum, the three bodies movetoward/away from their CM along straight lines. These implosion/explosion solutions arecalled Lagrange homotheties. Euler collinear and Lagrange equilateral configurations are theonly central configurations in the three-body problem. In 1912, Karl Sundmann showedthat triple collisions are asymptotically central configurations.Can planets collide, be ejected from the solar system or suffer significant deviations fromtheir Keplerian orbits? This is the question of the stability of the solar system. In the 18 th century, Lagrange and Pierre-Simon Laplace obtained the first significant results on stability.They showed that to first order in the ratio of planetary to solar masses ( M p /M S ), thereis no unbounded variation in the semi-major axes of the orbits, indicating stability of thesolar system. Their compatriot Sim´eon Denis Poisson extended this result to second orderin M p /M S . However, in what came as a surprise, Spiru Haretu (1878) overcame significanttechnical challenges to find secular terms (growing linearly and quadratically in time) in thesemi-major axes at third order! Haretu’s result did not prove instability as the effects of hissecular terms could cancel out. However, it effectively put an end to the hope of proving thestability/instability of the solar system using such a perturbative approach.The development of Hamilton’s mechanics and its refinement in the hands of Carl Jacobiwas still fresh when the dynamical astronomer Charles Delaunay (1846) began the firstextensive use of canonical transformations in perturbation theory [32]. The scale of hishand calculations is staggering: he applied a succession of 505 canonical transformations toa 7 th order perturbative treatment of the three-dimensional elliptical restricted three-bodyproblem . He arrived at the equation of motion for the small mass in Hamiltonian form using3 pairs of canonically conjugate orbital variables (3 angular momentum components, the trueanomaly, longitude of the ascending node and distance of the ascending node from perigee). Three-body configurations in which the acceleration of each particle points towards the CM and is pro-portional to its distance from the CM ( a b = ω ( R CM − r b ) for b = 1 , , 3) are called ‘central configurations’. The restricted three-body problem is a simplified version of the three-body problem where one of themasses is assumed much smaller than the two primaries. .1. SOME LANDMARKS IN THE HISTORY OF THE THREE–BODY PROBLEM . He found a new family of periodic orbitsin the circular restricted (Sun-Earth-Moon) three-body problem by using a frame rotatingwith the Sun’s angular velocity instead of that of the Moon. The solar perturbation to lunarmotion around the Earth results in differential equations with periodic coefficients. He usedFourier series to convert these ODEs to an infinite system of linear algebraic equations anddeveloped a theory of infinite determinants to solve them and obtain a rapidly convergingseries solution for lunar motion. He also discovered new ‘tight binary’ solutions to the three-body problem where two nearby masses are in nearly circular orbits around their center ofmass CM , while CM and the far away third mass in turn orbit each other in nearlycircular trajectories.The mathematician/physicist/engineer Henri Poincar´e began by developing a qualitativetheory of differential equations from a global geometric viewpoint of the dynamics on phasespace. This included a classification of the types of equilibria on the phase plane (nodes,saddles, foci/spiral and centers). His 1890 memoir on the three-body problem was the prize-winning entry in King Oscar II’s 60 th birthday competition (for a detailed account see [5]). Heproved the divergence of series solutions for the three-body problem developed by Delaunay,Hugo Gyld´en and Lindstedt (in many cases) and convergence of Hill’s infinite determinants.To investigate the stability of three-body motions, Poincar´e defined his ‘surfaces of section’and a discrete-time dynamics via the ‘return map’ (see Fig. A.2a). A Poincar´e surface S is atwo-dimensional surface in phase space transversal to trajectories. The first return map takesa point q on S to q , which is the next intersection of the trajectory through q with S .Given a hyperbolic fixed point (e.g., a saddle point) p on a surface S , he defined its stableand unstable spaces W s and W u as points on S that tend to p upon repeated forward orbackward applications of the return map (see Fig. A.2b). He initially assumed that W s and W u on a surface could not intersect and used this to argue that the solar system is stable.This assumption turned out to be false, as he discovered with the help of Lars Phragm´en. Simon Newcomb’s project of revising all the orbital data in the solar system established the missing42 (cid:48)(cid:48) in the 566 (cid:48)(cid:48) centennial precession of Mercury’s perihelion. This played an important role in validatingEinstein’s general theory of relativity. APPENDIX A. THREE–BODY PROBLEM q q S (a) h -1 h h W u W s p (b) Figure A.2: (a) A Poincare surface S transversal to a trajectory is shown. The trajectory through q on S intersects S again at q . The map taking q to q is called Poincar´e’s first return map.(b) The saddle point p and its stable and unstable spaces W s and W u are shown on a Poincar´esurface through p . The points at which W s and W u intersect are called homoclinic points, e.g., h , h and h − . Points on W s (or W u ) remain on W s (or W u ) under forward and backwarditerations of the return map. Thus, the forward and backward images of a homoclinic point underthe return map are also homoclinic points. In the figure, h is a homoclinic point whose image is h on the segment [ h , p ] of W s . Thus, W u must fold back to intersect W s at h . Similarly, if h − is the backward image of h on W u , then W s must fold back to intersect W u at h − . Furtheriterations produce an infinite number of homoclinic points accumulating at p . The first exampleof a homoclinic tangle was discovered by Poincar´e in the restricted three-body problem and is asignature of its chaotic nature. In fact, W s and W u can intersect transversally on a surface at a homoclinic point if thestate space of the underlying continuous dynamics is at least three-dimensional. What ismore, he showed that if there is one homoclinic point, then there must be infinitely many ofthem accumulating at p (see Fig. A.2b). Moreover, W s and W u fold and intersect in a verycomplicated ‘homoclinic tangle’ in the vicinity of p . This was the first example of what wenow call chaos.When two gravitating point masses collide, their relative speed diverges and solutions tothe equations of motion become singular at the collision time t c . More generally, a singularityoccurs when either a position or velocity diverges in finite time. Paul Painlev´e (1895) showedthat binary and triple collisions are the only possible singularities in the three-body problem.However, he conjectured that non-collisional singularities (e.g. where the separation betweena pair of bodies goes to infinity in finite time) are possible for four or more bodies. It tooknearly a century for this conjecture to be proven, culminating in the work of Donald Saari andZhihong Xia (1992) and Joseph Gerver (1991) who found explicit examples of non-collisionalsingularities in the 5-body and 3 n -body problems for n sufficiently large [71]. In Xia’sexample, a particle oscillates with ever-growing frequency and amplitude between two pairs Homoclinic refers to the property of being ‘inclined’ both forward and backward in time to the samepoint. .1. SOME LANDMARKS IN THE HISTORY OF THE THREE–BODY PROBLEM x = − k/x is singular at the collisionpoint x = 0. This singularity can be regularized by introducing a new coordinate x = u and a reparametrized time ds = dt/u , which satisfy the nonsingular oscillator equation u (cid:48)(cid:48) ( s ) = Eu/ E = (2 ˙ u − k ) /u . Such regularizations could shedlight on near-collisional trajectories (‘near misses’) provided the differential equations remainphysically valid .Karl Sundman (1912) began by showing that binary collisional singularities in the three-body problem could be regularized by a repararmetrization of time, s = | t − t | / where t isthe binary collision time [73]. He used this to find a convergent series representation (in powersof s ) of the general solution of the three-body problem in the absence of triple collisions . Thepossibility of such a convergent series had been anticipated by Karl Weierstrass in proposingthe three-body problem for King Oscar’s 60th birthday competition. However, Sundman’sseries converges exceptionally slowly and has not been of much practical or qualitative use.The advent of computers in the 20 th century allowed numerical investigations into thethree-body (and more generally the n -body) problem. Such numerical simulations have madepossible the accurate placement of satellites in near-Earth orbits as well as our missions tothe Moon, Mars and the outer planets. They have also facilitated theoretical explorationsof the three-body problem including chaotic behavior, the possibility for ejection of onebody at high velocity (seen in hypervelocity stars [8]) and quite remarkably, the discoveryof new periodic solutions. For instance, in 1993, Chris Moore discovered the zero angularmomentum figure-8 ‘choreography’ solution. It is a stable periodic solution with bodies ofequal masses chasing each other on an ∞ -shaped trajectory while separated equally in time(see Fig. A.3b). Alain Chenciner and Richard Montgomery [14] proved its existence usingan elegant geometric reformulation of Newtonian dynamics that relies on the variationalprinciple of Euler and Maupertuis. In fact, we use the associated Jacobi-Maupertuis metricformulation in our geometric approach to the planar three-body problem in Chapter 2. Solutions which could be smoothly extended beyond collision time (e.g., the bodies elastically collide)were called regularizable. Those that could not were said to have an essential or transcendent singularity atthe collision. Note that the point particle approximation to the equations for celestial bodies of non-zero size breaksdown due to tidal effects when the bodies get very close Sundman showed that for non-zero angular momentum, there are no triple collisions in the three-bodyproblem. APPENDIX A. THREE–BODY PROBLEM (a) Xia’s example m m m (b) Figure-8 solution Figure A.3: (a) An example due to Xia leading to a non-collisional singularity in the 5-body problemwhere a mass oscillates with ever-growing frequency and amplitude between two pairs of collapsingtight binaries that escape to infinity in finite time. (b) Equal-mass zero-angular momentum figure-8choreography solution to the three-body problem. A choreography is a periodic solution where allmasses traverse the same orbit separated equally in time. A.2 Proof of an upper bound for the scalar curvature Here we establish a strict lower bound on the quantity that appears in the relation (2.44)between Ricci scalars on C and S . Since Montgomery has shown that R S ≤ 0, this helpsus establish strictly negative upper bounds for the scalar curvatures on C , R and S . Wewill show here that 12 h + |∇ h | > ζh where ζ = 55 / ≈ . . (A.3)The best possible ζ is estimated numerically to be ζ = 8 / E , , . We define the power sum symmetric functions u n = (cid:80) i =1 v ni in termsof which the pre-factor in the JM metric (2.16) is h = v + v + v = u . In [55] Montgomeryshows that |∇ h | = 4 s where the symmetric polynomial s = (1 / (cid:0) − u + 4 u u − u + 3 u (cid:1) . (A.4)This gives12 h + |∇ h | = u (8 A + 6 B ) where A = u + u u and B = u − u u . (A.5)We will show below that A ≥ / 27 and B > − / 2, from which Eq. (A.3) follows (numeri-cally we find that B ≥ − / 81 which leads to the above-mentioned optimal value ζ = 8 / B , we define c = cos 2 η and s = sin 2 η cos 2 ξ which lie in theinterval [ − , u − u u > − ⇔ u − u + u > ⇔ (cid:0) − c + s ) − c + 24 cs (cid:1) > . (A.6) .2. PROOF OF AN UPPER BOUND FOR THE SCALAR CURVATURE − c + 24 cs > ≤ c ≤ 1. For − ≤ c < c = − d . Then it is enough to show that 17 + 8 d − d (1 − d ) > s ≤ − d . This holds as the LHS is positive at its boundary points d = 0 , d = 1 / A defined in Eq. (A.5) is a symmetric function of v , v and v which inturn are functions of η and ξ (2.16) for 0 ≤ η ≤ π/ ≤ ξ ≤ π . Since (cid:80) i /v i = 3,we may regard A as a function of any pair, say v and v . The allowed values of η and ξ define a domain ¯ D = D (cid:113) ∂D in the v - v plane. To show that A ≥ / 27, we seek its globalminimum, which must lie either at a local extremum in the interior D or on the boundary ∂D . ∂D is defined by the curves ξ = 0 and ξ = π/ η = 0 and η = π/ v = ∞ , v = 2 / 3) and ( v = 2 / , v = ∞ ) (see Fig. A.4). This isbecause, for any fixed η , v and v (2.16) are monotonic functions of ξ for 0 ≤ ξ ≤ π/ ξ = π/ 2. Along ∂D , A = (5 cos 6 η +22) / 27 is independentFigure A.4: The boundary ∂D of the region D in the v - v plane is given by the level curves ξ = 0 , π/ 2. These level curves run from the collision point η = 0 to the Euler point η = π/ v = ∞ or v = ∞ (where η = π/ ξ = π/ , π/ , π/ D are also shown. Note that D lies within the quadrant v , ≥ / of ξ and minimal at the Euler configurations η = π/ π/ / 27, which turns out to be the global minimum of A . This is because its only localextremum in D is at the Lagrange configuration v = v = v = 1 where A = 2 / 3. To seethis, we note that local extrema of A in D must lie at the intersections of ∂A/∂v = 0 and ∂A/∂v = 0. Now ∂A/∂v = ( v − v ) F ( v , v ) /v u where F ( v , v ) = u (cid:8) v + v + 2 (cid:0) v + v v + v (cid:1)(cid:9) − v + v )( u + u ) . (A.7)For ∂A/∂v to vanish, either v = v or F ( v , v ) = 0 or one of the v i = ∞ . The collisionpoints v i = ∞ do not lie in D . The conditions for ∂A/∂v to vanish are obtained via theexchange v ↔ v . The intersection of the conditions v = v and v = v lies at the Lagrange0 APPENDIX A. THREE–BODY PROBLEM configurations v i = 1 where A = 2 / 3. It turns out that the only intersection of v = v with F ( v , v ) = 0 or of v = v with F ( v , v ) = 0 lying in D occurs at the above Lagrangeconfiguration. For instance, when v = v = v , F ( v , v ) = − v (4 v − v − / (3 v − vanishes when v = 1 or v = 1 / v ≥ / F ( v , v ) and F ( v , v ), which using u − = 3, must satisfy F ( v , v ) − F ( v , v ) = ( v − v ) [12 v v v − ( v + v + v )] = 0 . (A.8)So either v = v or 12 v v v = u . Now, we have shown above that the only extrema of A on v = v in D lie at the Lagrange configurations. Since A is a symmetric function of the v i , it follows that its only extrema on v = v also lies at the Lagrange configurations. Onthe other hand, 12 v v v − ( v + v + v ) ≥ v i ≥ / 2, with equality only at v i = 1 / D . Thus the only extremum of A in D is at the Lagrange configurations(where A = 2 / 3) and hence its global minimum occurs on ∂D at the Euler configurations(where A = 17 / ppendix BThree-rotor problem B.1 Quantum N -rotor problem from XY model The quantum N -rotor problem may be related to the 2d XY model of classical statisticalmechanics which displays the celebrated Kosterlitz-Thouless topological phase transition [72].The dynamical variables of the XY model are 2d unit-vector spins S α (or phases e iθ α ) ateach site α of an N × M rectangular lattice with horizontal and vertical spacings a and b and nearest neighbor ferromagnetic interaction energies − J S α · S β = − J cos( θ α − θ β ) with J > a = b and assumes that θ varies gradually so thatin the continuum limit a → N, M → ∞ holding aN and aM fixed, the Hamiltonianbecomes H = J (cid:82) |∇ θ | d r . This defines the 1+1 dimensional O (2) principal chiral model.Here, we approximately reformulate the XY model as an interacting quantum N -rotorproblem by taking a partial continuum limit in the vertical direction followed by a Wick rota-tion. The resulting quantum system has been used to model a 1d array of coupled Josephsonjunctions and is known to be related to the XY model in a Villain approximation [77, 78]. i, N j , τ , M 2D lattice with XY spins ab ( ) ( )( ) Figure B.1: The quantum N -rotor problem arises from a partial continuum limit of the Wick-rotated XY model of classical statistical mechanics. APPENDIX B. THREE–ROTOR PROBLEM With i and j labelling the columns and rows of the lattice, the XY model Hamiltonian is H = − J (cid:88) i,j [cos( θ i,j +1 − θ i,j ) + cos( θ i +1 ,j − θ i,j )] (B.1)with J > 0. In the first term, the sum is over 1 ≤ i ≤ N and 1 ≤ j ≤ M − ≤ i ≤ N − ≤ j ≤ M . We will impose periodicboundary conditions (BCs) in the horizontal but not in the vertical direction (open BCs arealso of interest). We will take a continuum limit in two steps. We first make the spacingbetween rows small by introducing a continuous vertical coordinate τ in place of j such that τ ( j + 1) − τ ( j ) = δτ = b . Next, we approximate cos( θ i,j +1 − θ i,j ) bycos( θ i ( τ + δτ ) − θ i ( τ )) ≈ − 12 ( θ i ( τ + δτ ) − θ i ( τ )) ≈ − θ (cid:48) i ( τ ) b dτ. (B.2)Here, we have chosen to write ( δτ ) as b dτ in anticipation of taking b → H = J (cid:88) i (cid:90) (cid:26) b θ (cid:48) i ( τ ) − b cos [ θ i +1 ( τ ) − θ i ( τ )] (cid:27) dτ (B.3)using the prescription (cid:80) j bf ( τ j ) → (cid:82) f ( τ ) dτ . The resulting partition function Z = (cid:90) N (cid:89) k =1 D [ θ k ] exp [ − βH ] (B.4)after a Wick rotation τ = ict , may be written as Z = (cid:90) D [ θ ] e iS/ (cid:126) where S (cid:126) = βJ c (cid:88) i (cid:90) dt (cid:20) b c ˙ θ i ( t ) + 1 b cos [ θ i +1 ( t ) − θ i ( t )] (cid:21) . (B.5)We introduced a parameter c > t has dimensions of time.We may take a second continuum limit, this time in the horizontal direction by replacing (cid:80) i by (cid:82) dxa by taking a → N → ∞ while holding aN and a/b fixed to get S (cid:126) ≈ βJ c (cid:90) dxa (cid:90) dt (cid:40) b c (cid:18) ∂θ∂t (cid:19) + 1 b cos (cid:18) a ∂θ∂x (cid:19)(cid:41) ≈ βJ c (cid:90) dx dt (cid:26) ba c ˙ θ − ab θ (cid:48) (cid:27) . (B.6)The path integral (cid:82) D [ θ ] e iS/ (cid:126) is what we would have obtained if we had taken the conventionalcontinuum limit ( a, b → 0) of the XY model partition function and then performed a Wickrotation. Our two-step continuum limit has allowed us to approximately identify the quantum N -rotor problem (B.5) where b has not yet been taken to zero.For fixed N, a and b , the physical interpretation of (B.5) is facilitated by letting Lb/acβ play the role of (cid:126) where L is a length that remains finite in the limit a, b → L could be .2. POSITIVITY OF THE JM CURVATURE FOR ≤ E ≤ G (cid:126) has dimensions of action and tendsto 0 at low temperatures where quantum fluctuations in the Wick rotated theory should besmall. With this identification of (cid:126) , we read off the classical action S [ θ ] = (cid:88) i (cid:90) (cid:26) J Lb ac ˙ θ i + J La cos [ θ i − θ i +1 ] (cid:27) dt. (B.7)Letting m = J/c , r = (cid:112) Lb /a and g = J L/a , the corresponding Hamiltonian (with θ N +1 ≡ θ ) H = N (cid:88) i =1 (cid:26) mr ˙ θ i + g [1 − cos ( θ i − θ i +1 )] (cid:27) (B.8)describes the equal mass N -rotor problem. The rotor angles θ i parametrize N circles whoseproduct is the N -torus configuration space. Though the rotors are identical, each is as-sociated to a specific site and thus are distinguishable. In particular, the wavefunction ψ ( θ , θ , · · · θ N ) need not be symmetric or antisymmetric under exchanges. We may also vi-sualize the motion by identifying all the circles but allowing the rotors/particles to remembertheir order from the chain. So particles i and j interact only if i − j = ± 1. In particular,particles with coordinates θ and θ can freely ‘pass through’ each other! Furthermore, on ac-count of the potential, particles i and i + 1 can also cross without encountering singularities.Finally, we note that the quantum Hamiltonian corresponding to (B.7),ˆ H = (cid:88) i − (cid:126) mr ∂ ∂θ i − g cos( θ i − θ i +1 ) (B.9)has been used to model a 1d array of coupled Josephson junctions (see Fig. 1.1b) with thecapacitive charging and Josephson coupling energies given by E C = (cid:126) /mr = L/aβ J and E J = g = J L/a [77]. B.2 Positivity of the JM curvature for ≤ E ≤ g Here, we prove that for 0 ≤ E ≤ g , the JM curvature R of § E > V ) of the ϕ - ϕ configuration torus. It is negative outside and approaches ±∞ on the Hill boundary E = V . It is convenient to work in Jacobi coordinates ϕ ± = ( ϕ ± ϕ ) / § P = cos ϕ + and Q = cos ϕ − . In these variables, R = g N E ( P, Q ) mr ( E − V ) where N E = 5 + 2 Q − P Q + 8 P Q + (cid:20) Eg − (cid:21) (2 P + 2 P Q − . (B.10)Since E − V > N E ≥ N E ≥ E = 0 and 4 g and (b)for E = 0, N E vanishes only at the ground state G while for E = 4 g , it vanishes only at thesaddles D, with both G and the Ds lying on the Hill boundary. Since G is distinct from the4 APPENDIX B. THREE–ROTOR PROBLEM Ds, linearity of N E then implies that N E > < E < g . It onlyremains to prove (a) and (b).To proceed, we regard N E as a function on the [ − , × [ − , P Q -square. (i) When E = 0 , N has only one local extremum in the interior of the P Q -square at (0 , 0) where N (0 , 0) = 8 . On the boundaries of the P Q -square, N ( ± , Q ) = 2(1 ∓ Q ) ≥ N ( P, ± 1) = 2( P ∓ (5 ± P ) ≥ N vanishing only at (1 , 1) and ( − , − 1) both of which correspond to G. Thus, N ≥ E = 4 g ,the local extrema in the interior of the P Q -square are at (0 , 0) and ( ± , ∓ / / √ N g takes the values 0 and 40 / 27. On the boundaries of the P Q -square, N g ( ± , Q ) = 2(1 ± Q )(5 ± Q ) ≥ N g ( P, ± 1) = 2(1 ± P )(1 ± P + 4 P ) ≥ N g vanishing only at (1 , − 1) and ( − , E = 4 g , N g ≥ B.3 Measuring area of chaotic region on the ‘ ϕ = 0 ’Poincar´e surface To estimate the fraction of the area of the Hill region (at a given E ) occupied by the chaoticsections on the ‘ ϕ = 0’ Poincar´e surface, we need to assign an area to the correspondingscatter plot (e.g., see Fig 3.11a). We use the DelaunayMesh routine in Mathematica totriangulate the scatter plot so that every point in the chaotic region lies at the vertex ofone or more triangles (see Fig. B.2a). For such a triangulation and a given d > 0, the d -area of the chaotic region is defined as the sum of the areas of those triangles with maximaledge length ≤ d (accepted triangles in Fig. B.2a). Fig. B.2b shows that the area initiallygrows rapidly with d , and then saturates for a range of d . Our best estimate for the area ofthe chaotic region is obtained by picking d in this range. Increasing d beyond this admitstriangles that are outside the chaotic region. Increasing the number of points in the scatterplot (either by evolving each IC for a longer time or by including more chaotic ICs, whichis computationally more efficient) reduces errors and decreases the threshold value of d asillustrated in Fig. B.2b. B.4 Power-law approach to ergodicity in time Assuming correlations decay sufficiently fast, as expected for a chaotic system, we give here aheuristic explanation for our observed (see § F ( p, ϕ ) be a dynamical .4. POWER–LAW APPROACH TO ERGODICITY IN TIME (a) (b) Figure B.2: (a) Accepted (chaotic, shaded lighter/blue) and rejected (regular, shaded darker/grey)triangles on Delaunay Mesh for a sample chaotic region on the ‘ ϕ = 0’ Poincar´e surface at E = 7for maximal edge length d = 1. The light colored region on the periphery inside the Hill regionconsists of regular sections. (b) Estimates of the fraction of chaos (area of accepted region/area ofHill region) for various choices of d . An optimal estimate for f is obtained by picking d where f saturates. The three data sets displayed have n = 1 , , t = 10 . variable with ensemble average at energy E denoted ¯ F = (cid:104) F (cid:105) e (3.64). Its time average, overthe interval [0 , T ], along an energy- E phase trajectory ( (cid:126)p i ( t ) , (cid:126)ϕ i ( t )) labelled i , is denoted˜ F i ( T ) = 1 T (cid:90) T F i ( t ) dt ≡ T (cid:90) T F ( (cid:126)p i ( t ) , (cid:126)ϕ i ( t )) dt. (B.13)To examine the rate at which time averages along different trajectories i approach the en-semble average, we define the mean square deviation of ˜ F i ( T ) from ¯ F for a family I oftrajectories: var F ( T ) = (cid:28)(cid:16) ˜ F i ( T ) − ¯ F (cid:17) (cid:29) ≡ I ) (cid:88) i ∈I (cid:16) ˜ F i ( T ) − ¯ F (cid:17) . (B.14)Expanding, we write the mean square deviation asvar F ( T ) = (cid:68) ˜ F i ( T ) (cid:69) + ¯ F − F (cid:68) ˜ F i ( T ) (cid:69) . (B.15)We now assume that the ICs for the trajectories in I are distributed uniformly with respectto the Liouville measure on the energy- E hypersurface. Since the dynamics is Hamiltonian,by Liouville’s theorem the trajectories remain uniformly distributed at all times T , so thatas I ) → ∞ , (cid:68) ˜ F i ( T ) (cid:69) = ¯ F . (B.16)Thus, the mean square deviation becomesvar F ( T ) = (cid:68) ˜ F i ( T ) (cid:69) − ¯ F = (cid:68) ˜ F i ( T ) − ¯ F (cid:69) APPENDIX B. THREE–ROTOR PROBLEM = (cid:28) T (cid:90) T (cid:90) T [ F i ( t ) F i ( t ) − ¯ F ] dt dt (cid:29) = 1 T (cid:90) T (cid:90) T (cid:10) F i ( t ) F i ( t ) − ¯ F (cid:11) dt dt . (B.17)We now assume that F i ( t ) and F i ( t ) are practically uncorrelated if | t − t | > (cid:15) for sometime (cid:15) , i.e., (cid:10) F ( t ) F ( t ) − ¯ F (cid:11) ≈ (cid:40) | t − t | > (cid:15) and C ( t − t ) otherwise (B.18)by time-translation invariance, for some (2nd cumulant) function C ( t − t ). We now changeintegration variables from t , to u = t − t and v = ( t + t ) / dt dt = du dv andassume T (cid:29) (cid:15) to getvar F ( T ) ≈ T (cid:90) T dv (cid:90) (cid:15) − (cid:15) du C ( u ) = 1 T (cid:90) (cid:15) − (cid:15) C ( u ) du. (B.19)Thus, the RMS deviation of time averages from the ensemble average vanishes like 1 / √ T as T → ∞ . ibliography [1] Altmann, E. G., Silva, E. C. da and Caldas, I. L., Recurrence Time Statistics for Finite SizeIntervals , Chaos, (4), 975 (2004).[2] Arnold, V. I., Mathematical Methods of Classical Mechanics , 2nd ed., Springer Verlag (1989),page 245.[3] Arnold, V. I. and Avez, A., Ergodic Problems of Classical Mechanics , W. A. Benjamin, NewYork (1968).[4] Balakrishnan, V., Nicolis, G. and Nicolis, C., Recurrence time statistics in deterministic andstochastic dynamical systems in continuous time: A comparison , Phys. Rev. E, (3), 2490(2000).[5] Barrow-Green, J., Poincar´e and the Three Body Problem , Amer. Math. Soc., Providence,Rhode Island (1997).[6] Brack, M., Mehta, M. and Tanaka, K., Occurrence of periodic Lam´e functions at bifurcationsin chaotic Hamiltonian systems , Journal of Physics A, , 8199 (2001).[7] Braun, O. M. and Kivshar, Y. S., Nonlinear dynamics of the Frenkel-Kontorova model ,Physics Reports, , 1-108 (1998).[8] Brown, W. R., Hypervelocity Stars in the Milky Way , Physics Today, (6), 52 (2016).[9] Calogero, F., Solution of a three-body problem in one dimension , J. Math. Phys., , 2191(1969).[10] Casetti, L., Pettini, M. and Cohen, E. G. D., Geometric approach to Hamiltonian dynamicsand statistical mechanics , Physics Reports, , 237 (2000). arXiv:cond-mat/9912092.[11] Celletti, A., Basics of Regularization Theory , in Chaotic Worlds: From Order to Disorder inGravitational N-Body Dynamical Systems , Eds. Steves, B. A., Maciejewski, A. J. and Hendry,M., 203-230, Springer Netherlands (2006).[12] Cerruti-Sola, M., Ciraolo, G., Franzosi, R. and Pettini, M., Riemannian geometry of Hamil-tonian chaos: Hints for a general theory , Phys. Rev. E, , 046205 (2008).[13] Cerruti-Sola, M. and Pettini M., Geometric description of chaos in two-degrees-of-freedomHamiltonian systems , Phys. Rev. E, , 179 (1996). BIBLIOGRAPHY [14] Chenciner, A. and Montgomery, R., A remarkable periodic solution of the three-body problemin the case of equal masses , Ann. of Math. Second Series, (3), 881-901 (2000).[15] Chenciner, A., Action minimizing solutions of the Newtonian n-body problem: from homologyto symmetry , Proceedings of the ICM, Beijing, , 255-264, World Scientific (2002).[16] Chenciner, A., Three Body Problem , Scholarpedia, (10):2111 (2007).[17] Chenciner, A., Poincar´e and the Three-Body Problem in Henri Poincar´e, 1912-2012, Poincar´eSeminar 2012 , Eds. Duplantier B. and Rivasseau V., Springer, Basel (2015).[18] Chirikov, B. V., Resonance processes in magnetic traps , At. Energ., , 630 (1959) [Englishtranslation: J. Nucl. Energy Part C: Plasma Phys., , 253 (1960)].[19] Chirikov, B. V., A universal instability of many-dimensional oscillator systems , Phys. Rep., , 263 (1979).[20] Choi, M. Y. and Thouless, D. J., Topological interpretation of subharmonic mode locking incoupled oscillators with inertia , Phys. Rev. B, , 014305 (2001).[21] Churchill, R. C., Pecelli, G. and Rod, D. L., Stability transitions for periodic orbits in Hamil-tonian systems , Arch. Rational Mech. Anal., (4), 313-347 (1980).[22] Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations , McGraw-Hill, New York (1955), pg 15.[23] Cuneo, N., Eckmann, J.-P. and Poquet, C., Non-equilibrium steady state and subgeometricergodicity for a chain of three coupled rotors , Nonlinearity, , 2397-2421 (2015).[24] Dechant, A., Lutz, E., Kessler, D. A. and Barkai, E., Fluctuations of Time Averages forLangevin Dynamics in a Binding Force Field , Phys. Rev. Lett., , 240603 (2011).[25] Diacu F. and Holmes P., Celestial Encounters: The Origins of Chaos and Stability , PrincetonUniversity Press, New Jersey (1996).[26] do Carmo, M., Riemannian Geometry , Birkh¨auser, Basel (1992).[27] Efimov, V., Energy levels arising from resonant two-body forces in a three-body system , PhysicsLetters B, , 563 (1970).[28] Feigenbaum, M. J., Quantitative universality for a class of nonlinear transformations , J. Stat.Phys., (1), 25 (1978).[29] Gao, J. B., Recurrence Time Statistics for Chaotic Systems and Their Applications , Phys.Rev. Lett., (16), 3178 (1999).[30] Giorgilli, A. and Lazutkin, V. F., Some remarks on the problem of ergodicity of the StandardMap , Phys. Lett. A, (5-6), 359 (2000).[31] Gutzwiller, M. C., Chaos in Classical and Quantum mechanics , Vol. 1, InterdisciplinaryApplied Mathematics, Springer-Verlag, New York (1990). IBLIOGRAPHY [32] Gutzwiller, M. C., Moon-Earth-Sun: The oldest three-body problem , Reviews of ModernPhysics, , 589 (1998).[33] Greene, J. M., Method for determining a stochastic transition , J. Math. Phys., , 1183(1979).[34] Hadjidemetriou, J., Periodic orbits in gravitational systems , In Chaotic Worlds: From Orderto Disorder in Gravitational N-Body Dynamical Systems, Proceedings of the Advanced StudyInstitute, Steves, B. A., Maciejewski A. J. and Hendry M. (eds.), 43-79 (2006).[35] Hirata, M., Poisson law for Axiom A diffeomorphisms , Ergodic Theory and Dynamical Sys-tems, (3), 533 (1993).[36] Hirata, M. Dynamical Systems and Chaos , Vol. 1, p. 87, World Scientific, New Jersey (1995).[37] Kac, M., Probability and Related Topics in Physical Sciences , Interscience, London (1959).[38] Kac, M., On the notion of recurrence in discrete stochastic processes , Bull. Amer. Math. Soc., , 1002-1010 (1947).[39] Kraemer, T., Mark, M., Waldburger, P., Danzl, J. G., Chin, C., Engeser, B., Lange, A. D.,Pilch, K., Jaakkola, A., Nagerl, H.-C. and Grimm, R., Evidence for Efimov quantum statesin an ultracold gas of caesium atoms , Nature, , 315 (2006).[40] Krishnaswami, G. S. and Senapati, H., Curvature and geodesic instabilities in a geo-metrical approach to the planar three-body problem , J. Math. Phys., , 102901 (2016);arXiv:1606.05091.[41] Krishnaswami, G. S. and Senapati, H., An introduction to the classical three-body problem:From periodic solutions to instabilities and chaos , Resonance , 87-114 (2019).[42] Krishnaswami, G. S. and Senapati, H., Stability and chaos in the classical three rotor problem ,Indian Academy of Sciences Conference Series (1), 139-143 (2019); arXiv:1810.01317.[43] Krishnaswami, G. S. and Senapati, H., Classical three rotor problem: periodic solutions,stability and chaos , Chaos, (12), 123121 (2019); arXiv:1811.05807.[44] Krishnaswami, G. S. and Senapati, H., Ergodicity, mixing and recurrence in the three rotorproblem , Chaos, (4), 043112 (2020); arXiv:1910.04455.[45] Kumar, P. and Miller, B. N., Dynamics of Coulombic and Gravitational Periodic Systems ,Phys. Rev. E, , 040202 (2016).[46] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence , Springer-Verlag, New York(1984).[47] Lakshminarayan, A., Santhanam, M. S. and Sheorey V. B., Local Scaling in HomogeneousHamiltonian Systems , Phys. Rev. Letts., , 396 (1996).[48] Lanczos, C., The variational principles of mechanics , 4th ed. Dover (1970), page 139. BIBLIOGRAPHY [49] Laskar, J., Is the Solar System stable? Progress in Mathematical Physics, , 239-270 (2013).[50] Lenci, M., Aperiodic Lorentz gas: recurrence and ergodicity , Ergodic Theory and DynamicalSystems, (3), 869 (2003).[51] Melnikov, V. K., On the stability of the center for time periodic perturbations , Tr. Mosk. Mat.Obs., , 352 (1963) [English translation: Trans. Mosc. Math. Soc., , 1-56 (1963)].[52] Milnor, J., Morse Theory , Princeton University Press, New Jersey (1969).[53] Montgomery, R., The N-body problem, the braid group, and action-minimizing periodic solu-tions , Nonlinearity, , 363 (1998).[54] Montgomery, R., Infinitely Many Syzygies , Archives for Rational Mechanics, , 311 (2002).[55] Montgomery, R., Hyperbolic pants fit a three-body problem , Ergodic Theory and DynamicalSystems, , 921-947 (2005).[56] Montgomery, R., The zero angular momentum three-body problem: all but one solution hassyzygies , Ergodic Theory and Dynamical Systems, , 1933 (2007).[57] Montgomery, R., N-body choreographies , Scholarpedia, (11):10666 (2010).[58] Montgomery, R., The three-body problem and the shape sphere , The American MathematicalMonthly, , 299-321 (2015).[59] Moore, C., Braids in classical dynamics , Phys. Rev. Lett., , 3675 (1993).[60] Musielak, Z. E. and Quarles B., The three-body problem , Reports on Progress in Physics, ,6, 065901 (2014).[61] Nakahara, M., Geometry, Topology and Physics , 2nd ed., Taylor & Francis, London (2003).[62] Nicolis, C., Nicolis, G., Balakrishnan, V. and Theunissen, M., Recurrence time statistics inlow-dimensional dynamical systems , in Stochastic Dynamics, Eds. Schimansky-Geier, L. andP¨oschel T., Springer-Verlag, Berlin (1997).[63] O’Neill, B., Semi-Riemannian Geometry , Academic Press (1983).[64] Pettini, M., Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics ,Springer (2007), chapter 3.[65] Newton, P. K., The N -Vortex Problem: Analytical Techniques , Springer-Verlag, New York,(2001).[66] Rabouw, F. and Ruijgrok, Th. W., Three particles on a ring , Physica A, (3), 500-516(1981).[67] Rajeev, S. G., Geometry of the Motion of Ideal Fluids and Rigid Bodies , arXiv:0906.0184[math-ph] (2009). IBLIOGRAPHY [68] Rajeev, S. G., Advanced Mechanics: From Euler’s Determinism to Arnold’s Chaos , OxfordUniv Press (2013).[69] Ramasubramanian, K. and Sriram, M. S., Global geometric indicator of chaos and Lyapunovexponents in Hamiltonian systems , Phys. Rev. E, , 046207 (2001).[70] Routh, E. J., A Treatise on the Stability of a Given State of Motion: Particularly SteadyMotion , Macmillan (1877).[71] Saari, D. G. and Xia, Z., Off to infinity in finite time , Notices of the AMS, , 538 (1993).[72] Sachdev, S., Quantum phase transitions , Cambridge University Press, Cambridge (2008).[73] Siegel, C. L. and Moser, J. K., Lectures on Celestial Mechanics , Springer-Verlag, Berlin(1971), page 31.[74] Sinai, Ya. G., Geodesic flows on manifolds of negative constant curvature , Dokl. Akad. NaukSSSR, (4), 752 (1960) [English translation: Sov. Math. Dokl., , 335 (1960)].[75] Sinai, Ya. G., The central limit theorem for geodesic flows on manifolds of constant negativecurvature , Dokl. Akad. Nauk SSSR, (6), 1303 (1960) [English translation: Sov. Math.Dokl., , 983 (1960)].[76] Sinai, Ya. G., Dynamical Systems with Elastic Reflections , Russ. Math. Surv., , 137 (1970).[77] Sondhi, S. L., Girvin, S. M., Carini, J. P. and Shahar, D., Continuous quantum phase tran-sitions , Rev. Mod. Phys., (1), 315 (1997).[78] Wallin, M., Sørensen, E., Girvin, S. M. and Young, A. P., Superconductor-insulator transitionin two-dimensional dirty boson systems , Phys. Rev. B, , 12115 (1994).[79] Whittaker, E. T., A treatise on the analytical dynamics of particles & rigid bodies , 2nd Ed.,Cambridge University Press, Cambridge (1917).[80] Xia, Z., Arnold diffusion in the elliptic restricted three-body problem , J. Dyn. Diff. Equat., (2), 219 (1993).[81] Yeomans, D. K., Exposition of Sundman’s regularization of the three-body problem , NASAGoddard Space Flight Center Technical Report, NASA-TM-X-55636, X-640-66-481 (1966).[82] Yoshida, H., A Type of Second Order Ordinary Differential Equations with Periodic Coef-ficients for which the Characteristic Exponents have Exact Expressions , Celest. Mech., ,73-86 (1984).[83] Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport , Physics Reports, ,461 (2002).[84] Ziglin, S. L., Branching of solutions and nonexistence of first integrals in Hamiltonian me-chanics. I , Functional Analysis and Its Applications,16