Classical n -body system in geometrical and volume variables. I. Three-body case
A. M. Escobar-Ruiz, R. Linares, Alexander V Turbiner, Willard Miller Jr
CClassical n -body system in geometrical and volume variables. I. Three-body case A. M. Escobar-Ruiz, a) R. Linares, b) Alexander V Turbiner, c) and Willard Miller,Jr. d)1) Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa,San Rafael Atlixco 186, M´exico, CDMX, 09340 M´exico Instituto de Ciencias Nucleares, UNAM, M´exico DF 04510,Mexico School of Mathematics, University of Minnesota, Minneapolis MN 55455,U.S.A. a r X i v : . [ m a t h - ph ] A ug e consider the classical 3-body system with d degrees of freedom ( d >
1) at zerototal angular momentum. The study is restricted to potentials V that depend solelyon relative (mutual) distances r ij = | r i − r j | between bodies. Following the proposalby Lagrange, in the center-of-mass frame we introduce the relative distances (com-plemented by angles) as generalized coordinates and show that the kinetic energydoes not depend on d , confirming results by Murnaghan (1936) at d = 2 and vanKampen-Wintner (1937) at d = 3, where it corresponds to a 3D solid body. Realizing Z -symmetry ( r → − r ) we introduce new variables ρ = r , which allows us to makethe tensor of inertia non-singular. The 3 body positions form a triangle (of interac-tion) and the kinetic energy is S -permutationally invariant wrt interchange of bodypositions and masses (as well as wrt interchange of edges of the triangle and masses).For equal masses, we use lowest order symmetric polynomial invariants of Z ⊗ ⊕ S to define new generalized coordinates, they are called the geometrical variables . Twoof them of the lowest order (sum of squares of sides of triangle and square of thearea) are called volume variables . It is shown that for potentials, which depend ongeometrical variables only (i) and those which depend on mass-dependent volumevariables alone (ii), the Hamilton’s equations of motion look amazingly simple. Inthe case (ii) all trajectories are mass-independent!We study three examples in some detail: (I) 3-body Newton gravity in d = 3, (II)3-body choreography on the algebraic lemniscate by Fujiwara et al, and (III) the(an)harmonic oscillator.Keywords: Classical mechanics, 3-body system, Symmetries a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected] . INTRODUCTION The Lagrangian for a classical system of two non-relativistic particles with masses m and m , each of them with d degrees of freedom, has the form L = 12 m ˙ r + 12 m ˙ r − V ( | r − r | ) , (1)where r i ∈ R d denotes the position of the i th particle, ˙ r i ≡ ddt r i its velocity and V is atranslational invariant interaction potential with rotational symmetry. Solving this wellknown system is one of the basic problems in classical mechanics: it appears in manytextbooks. After separation of the d -dimensional center-of-mass (cms), the 2 d -dimensionalproblem is reduced to a d -dimensional one in the space of relative motion . In the particularcase of zero total angular momentum the number of degrees of freedom in the Lagrangian(1) is reduced to one. Thus, the relative motion is described by the Lagrangian˜ L = 12 µ ˙ r − V ( r ) , (2)where µ = m m m + m is the reduced mass of the system. The dynamical variable r ≡| r − r | has an elementary geometrical interpretation : it is the length of the interval which connectstwo particles. We call this the interval of interaction .In the case of three bodies with d -degrees of freedom, d ≥
1, the number of relativedistances is equal to the number of edges of the triangle which is formed by taking thebodies positions as vertices. This triangle defines the natural geometrical structure in whichthe interaction of the bodies occurs. We call it the triangle of interaction . In 1936, usingthe 3 mutual distances r ij (edges of the triangle of interaction) as generalized coordinates( r − representation), Murnaghan introduced a canonical transformation to reduce the pla-nar problem ( d = 2) from the configuration space R to R . It implies that the originalconfiguration space R is decomposed into the product R × R × S where the first factorcorresponds to the planar cms motion whilst the last one corresponds to the cyclic (angular)motion of the system around the center-of-mass. Therefore, the problem is formulated inthe three-dimensional space R ⊂ R of the relative motion. The natural extension of thisreduction to the case of 3-degrees of freedom ( d = 3) was then elaborated in Ref. .It follows from Ref. and Ref. that at zero total angular momentum, for both two andthree degrees of freedom ( d = 2 ,
3) the space of relative motion is the same R and the3orresponding reduced Hamiltonians defined in a six-dimensional phase space R × R coincide. It will be shown in this paper that it remains true for any number of degrees offreedom, d >
1. Let us denote this reduced Hamiltonian as H . It has the form of thekinetic energy of the 3D solid body with a certain tensor of inertia plus external potential.Following identification of the coefficients of the tensor of inertia in H as the entries ofa contravariant metric (cometric), the emerging Hamiltonian describes a three-dimensionalparticle moving in a curved space with cometric g µ ν ( r ij ). Remarkably, the components of g µ ν ( r ij ) are rational functions of r ij as well as its determinant ? .The triangle of interaction is characterized by three edges r ij (intervals of interaction).It is easy to see that the free Hamiltonian H , i.e. when the potential V is absent, is Z ⊗ -invariant under the reflections r ij → − r ij . It seems natural to make symmetry reductionintroducing Z -invariant coordinates ρ ij = r ij with their corresponding canonical momentum(it will be called ρ − representation). In these coordinates, the free Hamiltonian H is asecond degree polynomial in momentum variables with linear ρ -dependent coefficients on thephase space . In the case of equal masses the free Hamiltonian H has extra permutationalsymmetry S with respect to permutations of the intervals of interaction r ij (or their squares ρ ij ). It suggests the introduction of new S -permutationally invariant coordinates in ρ -space σ = ρ + ρ + ρ ,σ = ρ ρ + ρ ρ + ρ ρ ,σ = ρ ρ ρ . (3)These coordinates, as well as ones of the original 3-body system, are invariant under thepermutations of the bodies. In variables (3), the free Hamiltonian H remains a polynomialin phase space ( σ, p σ ), see Ref. . Following the Cayley-Menger formula , instead of σ one can introduce the square of the area of the triangle of interaction, S = (4 σ − σ ).Making a canonical transformation one can show that the free Hamiltonian H in coordinates( P ≡ σ , S, T ≡ σ ) remains a polynomial. The variables ( P, S ) are called the volumevariables . The key task of this article is to derive the explicit form of the free Hamiltonian H that answers the question: What is the form of the reduced Hamiltonian in variables ( P, S, T ) ? The goal of the present study is to write, for the 3-body system with zero total angularmomentum and arbitrary d >
1, the reduced Hamiltonian H in the ρ − representation and4lso in the ( P, S, T )-representation. The motivation of the present paper is three-fold. Ourfirst aim is to demonstrate that the choice of the intervals of interaction squared , ρ ij = r ij , asdynamical variables leads to a deep connection between the geometrical characteristics of thetriangle of interaction and the dynamics of the system. Accordingly, it will be shown thatfor arbitrary masses the corresponding Hamiltonian in ρ variables describes a particle in acurved space with a certain (essentially non-flat) metric g µ ν ( ρ ). Unlike the r − representation,now the components of g µ ν ( ρ ) are first degree polynomials in ρ -variables and its determinantDet[ g µ ν ( ρ )] is proportional to the square of the area of the triangle of interaction.Secondly, just as for the case of three identical particles m = m = m = 1 we will showthat the set of dynamical coordinates ( P, S, T ) exhibit outstanding properties: • As mentioned before, they characterize geometrically the triangle of interaction andalso lead to a reduced Hamiltonian that describes a three-dimensional particle movingin a curved space with a d -independent metric g µ ν ( P, S, T ) (components of which arepolynomials). Unlike the quantum case , no effective potential appears. • The determinant Det (cid:2) g µ ν ( P, S, T ) (cid:3) is of definite sign, it vanishes when the triangle ofinteraction is isosceles. • The Det (cid:2) g µ ν ( P, S, T ) (cid:3) is, in fact, the discriminant of the fourth degree polynomialequation that defines the physically relevant 3-body Newtonian gravity potential at d = 3 in terms of the aforementioned variables. • Remarkably, for planar 3-body choreographic motion on an algebraic lemniscate byJacob Bernoulli, studied by Fujiwara et al. , two of the variables P, T become particular constants of motion and the trajectory is an elliptic curve.Thirdly, we investigate how the volume variables
P, S are modified in the case of arbitrarymasses. Assuming that the potential V depends on above-introduced volume variables( P, S ), we will show that for the original Hamiltonian there exist trajectories which dependon these two volume variables alone. In the present paper the classical counterpart of the3-body quantum problem, which was treated by some of the present authors in Ref. , leadsto new insights into the connection between the geometrical properties of the triangle ofinteraction and dynamics of the 3-body system.5he structure of the article is organized very simply. In Section II, we first review thesymmetric reduction of the planar 3-body system. Separating the cms motion and usingthe conserved total angular-momentum p Ω , the problem is reduced to one of three degreesof freedom in which the coordinates are the lengths r , r , r of the sides of the triangleof interaction. It is shown that the corresponding reduced Hamiltonian H describes a 3-dimensional particle moving in a curved space. Next, we introduce the variables ρ ij = r ij ( ρ -representation) for which H becomes a polynomial at p Ω = 0. In Section III we provethat for zero total angular momentum and arbitrary dimension d > H . The 3-body chain of harmonic oscillatorsis briefly revisited to illustrate this representation. The case of three identical particles m = m = m = 1 is studied in Section IV. For H , the novel set of variables ( P, S, T ) isintroduced and the corresponding equations of motion are presented explicitly. The 3-bodyNewtonian gravity potential in d = 3 and planar choreographic trajectories on algebraiclemniscate ( d = 2) are used to exemplify the properties of this representation. Finally,in Section V in the case of arbitrary masses with a certain class of potentials a further reduction of the problem to one of two degrees of freedom is accomplished using modified,mass-dependent volume variables as dynamical coordinates. For conclusions and futureoutlook see Section VI. II. THREE-BODY SYSTEM: PLANAR CASE ( d = 2 ) We consider a planar ( d = 2) classical system of three interacting point-like particles withmasses m , m and m , respectively. The Lagrangian is of the form, L = T − V ( r , r , r ) , (4)where the kinetic energy is given by T = 12 m ˙ r + 12 m ˙ r + 12 m ˙ r , (5)here r i ∈ R is the vector position of the i th body, ˙ r i ≡ ddt r i and the potential V dependson the relative distances r ij ≡ | r i − r j | , L is rotationally symmetric. The configurationspace is six-dimensional R ( r ) × R ( r ) × R ( r ).The kinetic energy (5) can be expressed in terms of center-of-mass and the relative coor-dinates of the three bodies , T = 12 M ˙ Y + 12 µ ˙ r + 12 µ ˙ r + 12 µ ˙ r = 12 M ˙ Y + 12 µ ( ˙ r + r ˙ θ ) + 12 µ ( ˙ r + r ˙ θ ) + 12 µ ( ˙ r + r ˙ θ ) , (6)where M = m + m + m is the total mass, Y = m r + m r + m r M is the center-of-mass vector,and µ ij ≡ m i m j M denotes a reduced-like mass. In (6), we also introduce polar coordinates forrelative vectors, i.e. r ij ≡ ( r ij , θ ij ) in the space of relative motion.For future convenience, we select the center-of-mass system as the inertial frame, Y =0 , ˙ Y = 0. Eventually, in this inertial frame the system is characterized by four degrees offreedom. A. r -representation Now, following Ref. we consider the three relative distances r , r , r , and the angle Ω = θ + θ + θ , (7)as the four generalized coordinates of the Lagrangian (4), see Fig. 1.We find the angular relations˙ θ = 13 ( ˙Ω + ˙ φ − ˙ φ ) ; ˙ θ = 13 ( ˙Ω + ˙ φ − ˙ φ ) ; ˙ θ = 13 ( ˙Ω + ˙ φ − ˙ φ ) , (8)where φ , φ , φ are the interior angles (mod π ) of the triangle formed by the three particles,which are completely determined by its sides r ij through the law of cosines.Substituting (8) into the kinetic energy T (6) we arrive at T = R + R ˙Ω + R ˙Ω , (9)7 r r 𝛳 𝜙 𝜙 𝜙 Figure 1: Planar three-body system ( d = 2) in the space of relative motion. The relativevectors are not independent, they obey the constraint r + r + r = . The angles θ ij in r ij ≡ ( r ij , θ ij ) as well as the angles φ i between the vectors r ij are indicated.where the coefficients are given by the formulae18 R = 9 (cid:0) µ ˙ r + µ ˙ r + µ ˙ r (cid:1) + µ r ( ˙ φ − ˙ φ ) + µ r ( ˙ φ − ˙ φ ) + µ r ( ˙ φ − ˙ φ ) R = µ r ( ˙ φ − ˙ φ ) + µ r ( ˙ φ − ˙ φ ) + µ r ( ˙ φ − ˙ φ )18 R = µ r + µ r + µ r . (10)As mentioned above, the derivatives of the quantities φ , φ , φ are completely determinedby the relative distances r ij . The kinetic energy (9) does not depend on Ω (7). Therefore,the angle Ω is a cyclic variable and its canonical momentum p Ω ≡ ∂∂ ˙Ω L = ∂∂ ˙Ω T = R + 2 R ˙Ω= µ r ˙ θ + µ r ˙ θ + µ r ˙ θ , (11)is a constant of motion (saying differently, the first integral), ˙ p Ω = 0. From (11) it followsthat p Ω is nothing but the total angular momentum of the system about its center of mass.8n terms of p Ω , the kinetic energy (9) takes the form T = R + p Ω − R R . (12)The conserved quantity p Ω allows us to reduce the problem to one of three degrees of freedomin which the corresponding coordinates are the relative distances r , r and r
31 1 . To thisend, it is convenient to introduce the Routhian R defined by the Legendre transformation R ( r ij , ˙ r ij , p Ω ) = L − p Ω ˙Ω . (13)It is a function of the radial variables r ij , ˙ r ij and the integral p Ω only.
1. Hamiltonian for the reduced three-dimensional problem
In this Section we switch, for convenience, from the Lagrangian formalism to the Hamil-tonian one. From (13) it follows that the reduced Hamiltonian of the system is defined ina six-dimensional phase space where the dynamical variables are the relative distances r ij and its conjugate momentum variables p ≡ ∂ R ∂ ˙ r , p ≡ ∂ R ∂ ˙ r , p ≡ ∂ R ∂ ˙ r . In particular, for zero total angular momentum ? p Ω = 0, the momentum variable p reads p = 116 I S (cid:52) (cid:20) r (cid:2) ( µ µ + µ µ + µ µ ) r r + 4 µ S (cid:3) ˙ r − r r (cid:2) µ + µ ) µ (cid:0) r + r − r (cid:1) r + µ µ (cid:0)(cid:0) r − r (cid:1) − r (cid:1) (cid:3) ˙ r − r r (cid:2) µ + µ ) µ (cid:0) r − r + r (cid:1) r + µ µ (cid:0)(cid:0) r − r (cid:1) − r (cid:1) (cid:3) ˙ r (cid:21) , (14)here I ≡ µ r + µ r + µ r , (15)is the moment of inertia with respect to the center of mass, and S (cid:52) ≡ (cid:0) r r + 2 r r + 2 r r − r − r − r (cid:1) , (16)is the square of the area of the triangle formed by the three particles. It was called the triangle of interaction , see Fig. 2. By a cyclic arrangement of the labels (1 , ,
3) ( i.e. (2 , , , ,
2) ) in (14) we obtain the other two momenta p and p . The two geometrical9uantities P m ≡ m + m + m m m m I and S m ≡ m m m m + m + m S (cid:52) will play an important role in thepresent study. They will be called modified volume variables . r r r r r m mm r Figure 2: Triangle of interaction in d = 3: the individual coordinate vectors r i markpositions of vertices of the triangle with sides r ij . The center-of-mass (the barycenter ofthe triangle) is marked by a (blue) bubble.Eventually, we arrive to the reduced Hamiltonian H r = p ˙ r + p ˙ r + p ˙ r − R = 12 (cid:20) p m + p m + p m + r + r − r m r r p p + r + r − r m r r p p + r + r − r m r r p p (cid:21) + 23 p Ω S (cid:52) (cid:20) p m r − m r m m r r r + p m r − m r m m r r r + p m r − m r m m r r r (cid:21) + V eff , m ij ≡ m i m j m i + m j , (17)where V eff = V + p Ω (cid:20) m r + 1 m r + 1 m r − r m r r − r m r r − r m r r (cid:21) , is an effective potential. The angular momentum p Ω is a Liouville integral, it Poisson-commutes with the Hamiltonian (17), { p Ω , H r } = 0. At zero angular momentum p Ω = 0 thesecond term in kinetic energy vanishes as well as the second term in the effective potential10 eff . Explicitly, at p Ω = 0 the above Hamiltonian (17) becomes H r = 12 (cid:20) p m + p m + p m + r + r − r m r r p p + r + r − r m r r p p + r + r − r m r r p p (cid:21) + V ( r , r , r ) . (18)The way how the reduced Hamiltonian (18) is written we call the r − representation.
2. The metric g µν ( r )The associated cometric for (17) defined by coefficients in front of the quadratic terms invariables ( p , p , p ) is given by g µν ( r ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r + r − r m r r r + r − r m r r r + r − r m r r m r + r − r m r r r + r − r m r r r + r − r m r r m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)Its determinant D m = Det[ g µν ( r )] possesses the remarkable factorization property D m = ( m + m + m ) m m m I S (cid:52) r r r , thus, it is proportional to the moment of inertia I (15) and the area (squared) S (cid:52) (16) ofthe triangle of interaction. Remark.
The above determinant D m is a rational function in r -variables. It effectivelydepends on the three coordinates I , S (cid:52) and T ≡ r r r . Note that it vanishes, D m = 0,iff the area of the triangle of interaction is equal to zero or a triple-body collision occurs. Itis singular at the two-body collision point. B. ρ -representation The cometric (19) as well as the original kinetic energy (12) are invariant formally underreflections Z ⊕ Z ⊕ Z , r → − r , r → − r , r → − r , S -group action (permutations of the bodies). If we introduce new variables, ρ = r , ρ = r , ρ = r , (20)with the corresponding canonical momenta P = 12 r p , P = 12 r p , P = 12 r p , (21)we immediately arrive at the Z -symmetry reduced Hamiltonian H ρ = 2 (cid:20) ρ P m + ρ P m + ρ P m + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P (cid:21) + 43 p Ω S (cid:52) (cid:20) P m ρ − m ρ m m ρ ρ + P m ρ − m ρ m m ρ ρ + P m ρ − m ρ m m ρ ρ (cid:21) + V eff , (22)where V eff = V + p Ω (cid:20) m ρ + 1 m ρ + 1 m ρ − ρ m ρ ρ − ρ m ρ ρ − ρ m ρ ρ (cid:21) . The Hamiltonian (22) is written in what we call the ρ − representation (cf. (17)).
1. The metric g µν ( ρ )The associated contravariant metric for (22) defined by coefficients in front of thequadratic terms in momentum variables ( P , P , P ) g µν ( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (23)is linear in ρ -coordinates, cf.(19). Its determinant is D m = Det (cid:2) g µν ( ρ ) (cid:3) = 2 m + m + m m m m × ( m m ρ + m m ρ + m m ρ ) (cid:0) ρ ρ + 2 ρ ρ + 2 ρ ρ − ρ − ρ − ρ (cid:1) , (24)12nd it is positive definite. It is worth noting a remarkable factorization property of thedeterminant D m = 32 ( m + m + m ) m m m I S (cid:52) = 643 ( m + m + m ) m m m P m S m . Hence, D m is a polynomial function of the ρ -variables and it depends on two coordinatesalone, namely the product of I and S (cid:52) .In three dimensional ρ -space with metric (23) is not flat. In this case, the Ricci scalar Rstakes the form Rs( ρ ) = 9 P ( ρ )4096 ( m + m + m ) P m S m , (25)where P ( ρ ) = (cid:88) i + j + k =3 a ijk ρ i ρ j ρ k , is a polynomial of degree 3 in the ρ -variables (it contains cubic terms only) with mass-dependent coefficients a = 3 m m − m ( m + m ) m m + m (cid:0) m − m m + 3 m (cid:1) m m − m + m ) (cid:0) m + m (cid:1) m a = 2 (cid:0) m m + m − m − m m (cid:1) m + m m (cid:0) m − m m + 7 m (cid:1) m − m m m + 7 m m a = (cid:0) m m − m − m m + 7 m (cid:1) m − m m ( m + 3 m ) m + m m (8 m + 7 m ) m + 2 m m a = − m + m ) (cid:0) m + m (cid:1) m + m m (cid:0) m − m m + 3 m (cid:1) m − m m ( m + m ) m + 3 m m a = 7 m m − m m m + m (cid:0) m − m m + 11 m (cid:1) m m + 2 (cid:0) m + 4 m m − m m − m (cid:1) m = 2 m m − m (3 m + m ) m m + m (7 m + 8 m ) m m + (cid:0) m − m m + 11 m m − m (cid:1) m a = (cid:0) m m − m − m m + 3 m (cid:1) m − m m (cid:0) m + m m + 10 m (cid:1) m + m m (3 m − m ) m − m m a = (cid:0) m − m m + 11 m m − m (cid:1) m − m m (3 m + m ) m + m m (7 m + 8 m ) m + 2 m m a = 2 m m + m (8 m + 7 m ) m m − m ( m + 3 m ) m m + (cid:0) m m − m − m m + 7 m (cid:1) m . (26) C. Reduced Hamiltonian at zero angular momentum: ρ -representation For the special case of vanishing angular momentum p Ω = 0, the Hamiltonian (22) reducesto H = 2 (cid:20) ρ P m + ρ P m + ρ P m + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P (cid:21) + V ( ρ , ρ , ρ ) . (27)In this case, the coefficients in the kinetic energy are linear in the ρ -coordinates. This classicalHamiltonian is in complete agreement with the result obtained through the procedure of dequantization from the quantum one, see Ref. . In the next section the above result willbe generalized to the non planar case d >
2. Hereafter, the form (27) of the reducedHamiltonian will be used throughout the text. It is the central object to explore in thepresent consideration.
III. THREE-BODY SYSTEM: NON-PLANAR CASE d > A. ρ -representation The extension of the symmetry reduction originally proposed by Murnaghan for planarcase d = 2 to the non-planar case d = 3 (from a 18-dimensional phase space to 8-dimensionalone) is presented in Eq.(55 ) in Ref. . In this case, the reduced Hamiltonian depends on14ight dynamical coordinates, namely, the three mutual distances ( r , r , r ), an angularvariable ω and their associated canonical conjugate momenta. However, at zero angularmomentum the ω − dependent terms disappear and the system is described by the samereduced Hamiltonian (27). In the ρ -representation it can be proved that this result is validin any dimension d > Theorem 1
The − body system in d -dimensions ( d ≥ ) with potential of the form V = V ( ρ , ρ , ρ ) and zero angular momentum is described by a six-dimensional re-duced Hamiltonian H (in ρ -representation) H = T + V ( ρ , ρ , ρ ) , (28) where the kinetic energy T is polynomial in coordinates ρ ij ≡ r ij and their conjugate mo-menta P ij . Explicitly, T = 2 (cid:20) ρ P m + ρ P m + ρ P m + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P (cid:21) . (29) Proof : Take free 3-body Hamiltonian (kinetic energy) T = 12 m p + 12 m p + 12 m p . (30)( p i = m i ˙ r i ). Let us make a canonical transformation (change of variables in the phasespace) from ( r i , p i ) variables to the d − dimensional center-of-mass variable, the three coor-dinates (squares of mutual distances) ρ ij , a suitable set of (2 d −
3) angular variables andtheir corresponding canonical momenta. The center-of mass-variables can be separated outcompletely. The ρ -variables are given by ρ (cid:96)k = d (cid:88) s =1 ( x (cid:96),s − x k,s ) , ≤ (cid:96) < k ≤ . Thus, p x (cid:96),s = 2 (cid:88) h (cid:54) = (cid:96), ( x (cid:96),s − x h,s ) P ρ (cid:96)h + · · · , where the non-explicit terms are proportional to the momentum variables associated withthe remaining angular coordinates. At zero total angular momentum, such terms vanish.15ence, we can compute the coefficient of P ρ (cid:96)k in (30) for (cid:96) < k . It is2 (cid:18) m (cid:96) + 1 m k (cid:19) d (cid:88) s =1 ( x (cid:96),s − x k,s ) = 2 m (cid:96) + m k m (cid:96) m k ρ (cid:96)k . Similarly the coefficient of P ρ (cid:96)k P ρ (cid:96)k (cid:48) for k < k (cid:48) is4 m (cid:96) d (cid:88) s =1 ( x (cid:96),s − x k,s )( x (cid:96),s − x k (cid:48) ,s ) = 4 m (cid:96) ( r (cid:96) − r k ) · ( r (cid:96) − r k (cid:48) ) = 2 m (cid:96) ( ρ (cid:96)k + ρ (cid:96)k (cid:48) − ρ kk (cid:48) ) , where the last equality follows from the law of cosines. The coefficient of P ρ (cid:96)k P ρ (cid:96) (cid:48) k (cid:48) for k, k (cid:48) , (cid:96), (cid:96) (cid:48) all pairwise distinct is 0. At zero center-of-mass momentum (at rest frame) wearrive at T . (cid:50) Hence, for any dimension d > V . B. One center case: m → ∞ An interesting special case of the three-body system (27) emerges when m → ∞ andother two masses are kept finite. In general, the limit m → ∞ when keeping m , m finitecorresponds to the physical situation where one mass is much heavier than the others (forinstance, as in the earth-moon-sun system). We call this one center case . The Hamiltonian(27) becomes H ( m →∞ )0 = 2 (cid:20) ρ P m + ρ P m + ρ P m + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P (cid:21) + V ( ρ , ρ , ρ ) . (31)For the case m → ∞ , the associated contravariant metric for (31) g µν ( m →∞ ) ( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ m ρ + ρ − ρ m ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ + ρ − ρ m ρ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (32)16s linear in ρ -coordinates, and possesses a factorizable determinant D ( m →∞ ) = Det (cid:2) g µν ( m →∞ ) ( ρ ) (cid:3) = 32 m m (cid:18) ρ m + ρ m (cid:19) S (cid:52) , and it is positive definite. We emphasize that the Hamiltonian (31) is six-dimensional, allthree ρ -variables as well as their canonical momentum remain dynamical, see discussionbelow in Section III C. C. Two-center case: m , m → ∞ In the genuine two-center case two masses are considered infinitely heavy, m , m → ∞ ,thus, the reduced mass m also tends to infinity, while the third mass m = m remains finite.It implies that the coordinate ρ is unchanged in dynamics (being constant of motion in theprocess of evolution, it can be treated as external parameter), while two other ρ -variablesremain dynamical.The 3-body system is converted to a two-center problem. In the limit m , m → ∞ both 1st row and 1st column in (23) vanish as well as the determinant (24). However, theHamiltonian (27) in this limit is well-defined and finite. In particular, H ( m ,m →∞ )0 = 2 m (cid:20) ρ P + ρ P + ( ρ + ρ − ρ ) P P (cid:21) + V ( ρ , ρ , ρ ) . (33)This Hamiltonian (33) describes a two-dimensional particle moving in a curved space.From the coefficients in front of the second order terms, in momentum variables, in (33) onecan form the 2 × g µν ( m ,m →∞ ) = 1 m ρ ( ρ + ρ − ρ )( ρ + ρ − ρ ) 2 ρ , (34)with determinant D ( m ,m →∞ ) ≡ Det[ g µν ( m ,m →∞ ) ] = 16 m S (cid:52) , (35)cf. (24), (32), which remains positive definite.17 . Three-body closed chain of interactive harmonic oscillators As an application of the presented formalism, we consider the case of 3-body oscillatorwith quadratic potentials of interaction which depend on relative distances , | r i − r j | , only ? .Needless to say that the two-body harmonic oscillator can be reduced to a one-dimensionalradial Jacobi oscillator, see e.g. . In the 3-body case such a reduction is not possible ingeneral.The potential of a three-body closed chain of harmonic oscillators takes the form V (har) = 2 ω (cid:20) ν ρ + ν ρ + ν ρ (cid:21) , (36)where ω > ν , ν , ν ≥ H har0 = 2 (cid:20) ρ P m + ρ P m + ρ P m + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P + ρ + ρ − ρ m P P (cid:21) + 2 ω (cid:20) ν ρ + ν ρ + ν ρ (cid:21) , (37)cf.(27). Unlike the case of the standard three-dimensional harmonic oscillator the Hamilto-nian (37) is not superintegrable and it does not admit separation of variables. However, inthe special case m ν = m ν , m ν = m ν , m ν = m ν , (38)(any two relations imply that the third relation should hold), the system (37) admits 5 func-tionally independent constants of the motion, so it becomes maximally superintegrable . Ifone of the three constraints in (38) is fulfilled only the system becomes minimally superin-tegrable. IV. ( P, S, T ) -REPRESENTATION (GEOMETRICAL REPRESENTATION) In this Section, we consider the case of three equal masses m = m = m = 1. Byusing a canonical transformation, the geometrical properties of the triangle of interactionare translated to a certain dynamical variables (see below).18 . Reduced Hamiltonian Based on the Z ⊗ ⊕ S symmetry of the free Hamiltonian H (27), let us introduce itsinvariants as new variables( r , r , r ) → ( ρ , ρ , ρ ) → ( P, S, T ) , where P ≡ I = 12 ( ρ + ρ + ρ ) ,S ≡ S (cid:52) = 116 (cid:0) ρ ρ + 2 ρ ρ + 2 ρ ρ − ρ − ρ − ρ (cid:1) ,T ≡ ρ ρ ρ , (39)cf.(3). Their associated canonical momenta read P P ≡ ρ − ρ ) ( ρ − ρ ) ( ρ − ρ ) (cid:20) ρ ( ρ − ρ − ρ ) ( ρ − ρ ) P + ρ ( ρ − ρ ) ( ρ + ρ − ρ ) P + ρ ( ρ − ρ ) ( ρ − ρ + ρ ) P (cid:21) ,P S ≡ ρ ( ρ − ρ ) P + ρ ( ρ − ρ ) P + ρ ( ρ − ρ ) P )( ρ − ρ ) ( ρ − ρ ) ( ρ − ρ ) ,P T ≡ P − P ( ρ − ρ )( ρ − ρ ) + P − P ( ρ − ρ )( ρ − ρ ) , (40)respectively. In (39), the moment of inertia I ∝ P and the area (squared) S (cid:52) = S of thetriangle of interaction are translated to two new dynamical variables explicitly. The quanti-ties P and S were called volume variables . It is worth mentioning that each of the variables( P, S, T ) (39) is characterized by an accidental permutation symmetry S in ρ − coordinatesin addition to the symmetry S of interchange of any pair of bodies positions.In new variables the Hamiltonian H (27) takes the form H geo = 3 P P P + P S P S + T (cid:20) S + 4 P (cid:21) P T + 18 T P P P T + 12 S P P P S + (cid:20) S + 8 S P (cid:21) P T P S + V ( P, S, T ) . (41)It corresponds to 3D solid body motion in an external potential V . Due to the energyconservation H geo = E , the trajectories in the six-dimensional phase space are representedby the curves,3 P P P + P S P S + T (cid:20) S + 4 P (cid:21) P T + 18 T P P P T + 12 S P P P S + (cid:20) S + 8 S P (cid:21) P T P S + V ( P, S, T ) =
E . (42)19his Hamiltonian H geo can be interpreted as the way to describe a three-dimensional particlein a curved space, see below. Tensor of inertia can be identified with cometric in this case.The associated cometric for (41) is given by g µν ( P, S, T ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P S T S S P S (4 S + P )9 T S (4 S + P ) 4 (12 S + P ) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (43)Its components are polynomials in the volume variables. Its determinant admits factorizationto three factors, D geo ≡ Det (cid:2) g µν ( P, S, T ) (cid:3) = 3 P S (cid:16) P T (cid:0) S + P (cid:1) − S (cid:0) S + P (cid:1) − T (cid:17) . Now, let us study under what conditions the determinant vanishes, D geo = 0, so thatthe metric (tensor of inertia) becomes degenerate. In this case, the transformation (39)is singular (not-invertible). There are three possibilities. The first possibility is when thefirst factor vanishes, P = 0. We discard it, since it corresponds to the situation when theconfiguration space shrinks to a point (we call it triple collision point ). The second possibilityis S = 0 , (44)it corresponds to a collinear three-body configuration , or, equivalently, when three bodiesare on the line, the configuration space becomes two-dimensional, thus, it shrinks to plane,while the third possibility leads to,4 P T (cid:0) S + P (cid:1) − S (cid:0) S + P (cid:1) − T = 0 , (45)in which l.h.s. is homogeneous polynomial in ρ − coordinates. In order to solve (45) let ustake ρ ’s in the form ρ = A ρ , ρ = B ρ , ρ = C ρ , where
A, B, C are dimensionless parameters, and substitute them into
P, S and T as firstand then to (45). We arrive at( A − B )( B − C )( A − C ) = 0 . A = B and C is arbitrarynumber in the interval [0 , A ]. Hence, the determinant D geo vanishes for any isoscelestriangle of interaction, in particular, for equilateral triangle A = B = C , where12 S = P , T = 8 P . (46)The space with metric (43) is not flat. The Ricci scalar Rs geo can be calculated and ittakes the form Rs geo = P ( P, S, T ) (cid:0) D geo (cid:1) , (47)where P ( P, S, T ) is a certain polynomial of degree 8 in variables (
P, S, T ) being homoge-neous polynomial in ρ ’s of degree 18. B. Equations of motion
From the reduced Hamiltonian (41), we obtain the Newton equations of motion for thevariables (
P, S, T )¨ P = 32 D geo (cid:20) − S ˙ P (cid:16) S (cid:0) S + P (cid:1) − P T (cid:0) S + P (cid:1)(cid:17) +3 ˙ S T (cid:0) S P + 4 P − T (cid:1) − S ˙ T (cid:0) S − P (cid:1) − S ˙ S ˙ P T (cid:0) S − P (cid:1) +˙ T (cid:16) S ˙ P (cid:0) S (cid:0) S + P (cid:1) − T (cid:1) − S ˙ S (cid:0) S P + 2 P − T (cid:1)(cid:17) (cid:21) − T ∂ T V + 2 S ∂ S V + P ∂ P V ) , (48)¨ S = 32 D geo P (cid:20) ˙ S (cid:16) P T (cid:0) − S + 30 S P + P (cid:1) − S P (cid:0) S + P (cid:1) + 27 T (cid:0) S − P (cid:1)(cid:17) +6 S ˙ T (cid:0) S − P (cid:1) + 8 S ˙ P (cid:16) S (cid:0) S + P (cid:1) − P T (cid:0) S + P (cid:1)(cid:17) +2 S ˙ S ˙ P (cid:16) T (cid:0) S + 30 S P + P (cid:1) − S P (cid:0) S + P (cid:1) − P T (cid:17) +˙ T (cid:16) S ˙ S (cid:0) S P + 2 P − T (cid:1) − S ˙ P (cid:0) S (cid:0) S + P (cid:1) − P T (cid:1)(cid:17) (cid:21) − S (4 (cid:0) S + P (cid:1) ∂ T V + P ∂ S V + 6 ∂ P V ) , (49)21 T = 32 D geo P (cid:20) S ˙ P T (cid:0) S − S P + 14 S P − P T (cid:1) +˙ S T (cid:0) P (cid:0) S + 12 S P + P (cid:1) − P T (cid:0) S + P (cid:1) + 243 T (cid:1) + S ˙ T (cid:0) − S P + 12 S (cid:0) P + 9 T (cid:1) + 4 P − P T (cid:1) − S ˙ S ˙ P T (cid:0) S P − S (cid:0) P + 9 T (cid:1) − P + 45 P T (cid:1) + ˙ T (cid:18) S ˙ P (cid:0) S P − S (cid:0) P + 9 T (cid:1) − S P (cid:0) P − T (cid:1) + P T (cid:1) − S ˙ S (cid:0) − P T (cid:0) S + 13 P (cid:1) + 8 P (cid:0) S + P (cid:1) (cid:0) S + P (cid:1) + 81 T (cid:1) (cid:19) (cid:21) − T (cid:0) S + P (cid:1) ∂ T V + 4 S (cid:0) S + P (cid:1) ∂ S V + 9 T ∂ P V ) . (50)A further remark is in order. From the Hamiltonian (41) it follows that the time evolutionof P T (40) is given by the Hamilton’s equation˙ P T = − P T (cid:2) P P + 2 P T (12 S + P ) (cid:3) − ∂ T V . (51)Note when potential V depends on the volume variables P and S only, (51) is reduced to˙ P T = − P T (cid:2) P P + 2 P T (12 S + P ) (cid:3) , (52)which implies the existence of trajectories with P T = 0. In general, from the Hamilton’sequations ˙ P = ∂ H geo ∂P P , ˙ S = ∂ H geo ∂P S , ˙ T = ∂ H geo ∂P T , one can write the momentum variables ( P P , P S , P T ) in terms of velocities ( ˙ P , ˙ S, ˙ T ). Inparticular, the momentum P T is given by P T = 3 S D geo (cid:20) ˙ P (cid:0) S (cid:0) P + 4 S (cid:1) − P T (cid:1) + ˙ S (cid:0) T − P (cid:0) P + 4 S (cid:1)(cid:1) + ˙ T (cid:0) P − S (cid:1) (cid:21) . (53)Therefore, one can construct a reduced Hamiltonian on the volume variables phase space(
P, S, P P , P S ). This construction will be elaborated in the next Section.Now, within the ( P, S, T )-representation we consider two particular three-body exam-ples: (i) the R Newtonian gravity and (ii) the planar, R choreographic trajectories on thealgebraic lemniscate by Jacob Bernoulli (1694).22 . Three-body Newtonian gravity potential First, let us consider the three-body Newton problem in R ( d = 3). The potential in(41) reads V ≡ V γ = − γ (cid:18) r + 1 r + 1 r (cid:19) = − γ (cid:18) √ ρ + 1 √ ρ + 1 √ ρ (cid:19) , (54)where γ is the gravitational constant. In terms of the variables ( P, S, T ), see (39), one canshow that V γ is one of the roots of the fourth order algebraic equation T V γ − γ (4 S + P ) T V γ + 8 γ T V γ + γ (cid:18) (4 S + P ) − γ T P (cid:19) = 0 . (55)This equation is invariant wrt change ( γ → − γ ) and ( V γ → − V γ ) ? . Interestingly, theremaining three roots correspond to different 3-body Coulomb potentials for 3 unit chargesof different signs, V (1) γ = − γ (cid:18) − √ ρ + 1 √ ρ + 1 √ ρ (cid:19) V (2) γ = − γ (cid:18) √ ρ − √ ρ + 1 √ ρ (cid:19) V (3) γ = − γ (cid:18) √ ρ + 1 √ ρ − √ ρ (cid:19) . (56)Hence, the equation (55) describes all four possible 3-body Coulomb/Newton potentials withconstant of interaction γ .The discriminant of the equation (55) admits factorization D γ = 4096 γ T (cid:16) P T (cid:0) S + P (cid:1) − S (cid:0) S + P (cid:1) − T (cid:17) , (57)and its last factor coincides with the last factor in the factorized expression for determinant D geo of the metric (43). For any isosceles triangle both determinant and discriminant vanish, D γ = D geo = 0. Furthermore, in the case of an equilateral triangle of interaction thepotential (54) simplifies, V γ = − γ √ P .
It corresponds to the solution found long ago by Lagrange , it describes to the so called central configuration , see e.g. Refs. and reference therein. Also, the remarkable peri-odic Figure Eight solution of the three-body problem found by Moore and confirmed byChenciner and Montgomery - the so-called 3-body choreography - gets a natural presen-tation in the geometrical variables. On this trajectory, the motion of the bodies alternatesbetween six collinear configurations and six isosceles-triangular ones.23 . Choreographic motion As for the choreographic trajectories in R ( d = 2), when three bodies follow one to eachother being on the same curve, Fujiwara et al. solved the inverse problem of choreographicmotion on a Figure-8 given by the algebraic lemniscate of Jacob Bernoulli (1694) on ( x, y )-plane ( x + y ) = c ( x − y ) , (58)where without loss of generality one can put c = 1, and found 3-body pairwise potential.It was shown that such a figure eight is the choreographic trajectory for three unit mass,point-like particles with zero angular momentum with two-parametric potential V ∞ = 14 ln T − √ P , (59)with the volume variables
P, T given by (39), . The first attractive terms in (59) represents a3-body R Newtonian potential for gravitational constant γ = 1 /
2, while the second repulsiveterm is the square of the hyperradius in the space of relative motion. It is worth mentioningthat the Figure-8 algebraic lemniscate with potential (59) is close to the transcendentalFigure-8 trajectory found by C Moore , see also Ref. , for the R Newtonian gravitypotential.Recently, in Ref. it was found that the 3-body choreographic motion on the algebraiclemniscate is maximally (particularly) superintegrable. Moreover, the two variables P and T become particular integrals (see also Ref. ). Along this Figure-8 trajectory, these variablestake the constant value P = T = 3 √ . (60)Therefore, it is natural to work with the Hamiltonian (41). It can be shown that threeequations of motion (48)-(50) constrained by the conditions (60) lead to non-linear ODE S (cid:0) S + 864 S − (cid:1) + 48 √ S = 0 , (61)which defines the elliptic curve, see Fig.3, with variables P, T playing role of ellipticinvariants . The solution of (61) is given by the Weierstrass function ℘ ( t ; P, T ). Hence,in the configuration space parametrized by geometrical variables the 3-body choreographyon algebraic lemniscate by Jacob Bernoulli is given by a (planar) elliptic curve!24 t S Figure 3: Time evolution of the volume variable S , area (squared) of the triangle ofinteraction, for the 3-body choreographic Figure-8 on the algebraic lemniscate byBernoulli. The points S = 0 correspond to the so-called Euler line for which all threebodies are on the line. V. VOLUME VARIABLES REPRESENTATIONA. Case of equal masses
Now we consider the case of three unit masses m = m = m = 1 and focus on potentialsthat depend solely on the volume variables P and S , see (39), V = V ( P, S ) . (62)In this case the equations of motion (51) that emerge from the Hamiltonian H geo (41) admita number of solutions with P T = 0. This fact is not trivial since the momentum P T is not aLiouville integral of motion, { P T , H geo } P.B. (cid:54) = 0.Equivalently, in the six-dimensional phase space (
P, S, T, P P , P S , P T ) the hypersurface P T = c becomes an invariant manifold at c = 0: { P T , H geo } P.B. | PT =0 = 0. This impliesthat any trajectory of H geo for which the initial condition P T = 0 is imposed will remain onthe hypersurface H geo ( P, S, T, P P , P S , P T = 0) = E during the evolution. Inserting P T = 0directly into the Hamiltonian (41) we arrive at H vol ≡ H geo | PT =0 = 3 P P P + P S P S + 12 S P P P S + V ( P, S ) , (63)which describes a two-dimensional particle in a curve space, see below. Notice that T − dependence in (63) is absent as well. The corresponding Hamilton’s equations for25 vol are given by ˙ P = 6 P P P + 12 S P S ˙ P P = − P P − S P S − ∂ P V ˙ S = 2 P S P S + 12 S P P ˙ P S = − P P S − P P P S − ∂ S V , (64)hence, we arrive at the Newton equations¨ P = 12 D vol (cid:20) P S ˙ P − S ˙ P ˙ S + 9 P ˙ S (cid:21) − S ∂ S V + P ∂ P V )¨ S = 12 D vol (cid:20) S (cid:0) P − S (cid:1) − S ˙ P + 6 P S ˙ P ˙ S (cid:21) − S ( P ∂ S V + 6 ∂ P V ) , (65)cf.(48),(49). Remark.
It is worth clarifying the connection between the trajectories of H vol (63) andthose of H geo (41). The evolution P = P ( t ) and S = S ( t ) in (65) also satisfies the equationsof motion (48)-(50), where T = T ( t ) is fully determined by P ( t ) and S ( t ) if the condition P T = 0 is imposed, see (53). In this way we reduced effectively the dimensionality of thesystem (41) from six to four.The Hamiltonian (63) is written in what we call the volume variables representation . Themetric (or, equivalently, the tensor of inertia) for (63) is of the form g µν vol = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P S S P S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (66)with determinant D vol = 3 S ( P − S ) . In the case of an equilateral triangle of interaction there appears the relation P = 12 S and this determinant vanishes, D vol = 0. It also follows from (66) that the correspondingRicci scalar Rs is given by Rs vol = 3 S P ( S − D vol2 , (67)see below, (82). Rs vol is singular for equilateral triangle of interaction.An interesting particular case occurs when the potential (62) does not depend on S andis a function of the volume variable P alone V = V ( P ) .
26s it follows from (63) that there exist trajectories for the original Hamiltonian (41) whichdepend on P only. Such trajectories lie on the intersection between the hypersurfaces P T = 0and P S = 0, they are described by the two-dimensional Hamiltonian H vol | PS =0 ≡ H P = 3 P P P + V ( P ) , (68)cf.(63). The connection between the trajectories of H P (68) with those of H vol (63) and H geo (41) is the following. Taking a non-trivial solution P = P ( t ) (cid:54) = 0 of (68) one can define thefunction S = S ( t ) for which the canonical momentum P S vanishes in H vol . Explicitly, thecondition can be obtained from (63), P S = 2 ˙ P S − P ˙ S S (12 S − P ) = 0 . (69)Such P = P ( t ) and S = S ( t ) = P ( t ) satisfy ? the equations (65) for H vol . Moreover, theyalso obey the equations (48)-(50) for H geo where T turns out to be identically zero due tothe condition P T = 0, see (53). All that allows to reduce further the dimensionality of thesystem described by (41) from six to two.
1. Anharmonic Oscillator potential
As a concrete example, let us consider the following potential V ( AO ) = A P + B P = A ρ + ρ + ρ ) + B ρ + ρ + ρ ) , (70)where B ≥ A > B = 0. In terms of variable P the potential (70) corresponds toa shifted one-dimensional harmonic oscillator on the half line R + with non-standard kineticenergy. In the ρ − representation it corresponds to a certain three-dimensional quadratic po-tential on the first octant R whilst in the r − representation it describes a three-dimensionalisotropic harmonic oscillator with quartic anharmonicity ? . For the potential (70), theHamiltonian (68) becomes H ( AO ) P = 3 P P P + A P + B P , (71)and the corresponding equation of motion is given by¨ P = ( ˙ P ) P − P ( A + 2 B P ) . (72)27n new variable x = √ P , the equation (72) becomes¨ x = − A x − B x , (73)which describes a unit mass body moving on the half line R + in quartic potential v = ( A x + B x ). Its solution is known and P ( t ) = x = A k B (1 − k ) sn ( y, ik ) , y = ± (cid:114) A − k t , (74)( B (cid:54) = 0, k (cid:54) = ±
1) satisfies the equation (72), where sn( y, ik ) is the Jacobi elliptic functionwith imaginary elliptic modulus ( ik ). In the case of physical systems the function P ( t )should be positive and the parameter k (cid:54) = ± H ( AO ) P = A k B (1 − k ) . The dynamics of this system is presented in Fig.4. A = A = = P - - - (a) ( P P , P )-representation P - - - P (cid:1) A = A = = (b) ( ˙ P , P )-representation
Figure 4: Trajectories for the anharmonic oscillator (71): (a) in the phase space ( P P , P )and (b) in the physical space ( ˙ P , P ). They correspond to the energy H ( AO ) P = 1 and theparameters B = 1 and A = , , B = 0 in (71) we obtain the harmonic oscillator potential V = A P , which isquadratic polynomial in the r -representation. In this case the trajectories are trigonometricvibrations, P ( t ) = c cos ( √ A t + c ) , with energy H ( AO ) P = c A where c > , c are real constants of integration. In Fig.5 wepresent the phase portrait of the system as well. In general, due to the energy conservation28 ( AO ) P = E the trajectories in the phase space are always cubic curves,3 P P P + A P + B P = E , (see Figs.4a,5a). Note that at fixed values of the energy E and A the presence of anhar-monicity ( B >
0) in (71) tends to decrease the amplitude of the harmonic motion ( B = 0). A = A = = P - - P P (a) ( P P , P )-representation P - - - P A = A = = (b) ( ˙ P , P )-representation
Figure 5: Trajectories for the harmonic oscillator V = A P : (a) in the phase space( P P , P ) and (b) in the physical space ( ˙ P , P ). They correspond to the energy H ( AO ) P = 1and A = , , B. Case of unequal masses: modified volume variables
In this Section the case of three bodies with arbitrary masses ( m , m , m ) is consid-ered. From the general Hamiltonian H at zero angular momentum, see (27), it will beconstructed a four-dimensional reduced Hamiltonian, which depends on the modified vol-ume variables P m and S m alone and their respective canonical momenta, for two-variablepotentials V ( P m , S m ).As a first step, let us make the change of variables in the Hamiltonian H (27)( ρ , ρ , ρ ) → ( P m , S m , Q ) , (75)where the volume variable S m = 3 m m m m + m + m S , (76)is proportional to the square of the area of the triangle of interaction S (39), and P m = 12 (cid:20) m ρ + 1 m ρ + 1 m ρ (cid:21) , (77)29s the weighted sum of the edges (squared) of the triangle of interaction. At equal masses m = m = m = 1 the modified volume variables coincide to the original ones P and S : P m → P , S m → S . The third variable Q = Q ( ρ , ρ , ρ ) can be any function of ρ ’s withcondition that the Jacobian of the transformation (75) is invertible (nonsingular) in thedomain R , where the problem is defined. In these variables, the Hamiltonian (27) takesthe form H Q = m + m + m m m m P m P P m + P Q ( F P P m + F P S m + F P Q )+ m + m + m m m m P m S m P S m + 4 m + m + m m m m S m P P m P S m + V , (78)where, in general, the coefficients F , F and F are functions of ( P m , S m , Q ). Let us considerthat the family of two-variable potentials in (78) depends on the volume variables V = m + m + m m m m V m ( P m , S m ) , (79)alone. In this case it is easy to see that the Hamiltonian (78) admits trajectories with P Q = 0in the six-dimensional phase space ( P m , S m , Q, P P m , P S m , P Q ): { P Q , H Q } P.B. | PQ =0 = 0. Suchtrajectories are described by the Hamiltonian H m ≡ H | P Q =0 = m + m + m m m m (cid:20) P m P P m + P m S m P S m + 12 S m P P m P S m + V m ( P m , S m ) (cid:21) , (80)which generalizes H vol (63) to the non-equal mass case. Multiplying H m (80) by m m m m + m + m we obtain the Hamiltonian (63). Hence, we arrive to the interesting result The zero total angular momentum trajectories of a 3-body system with equalmasses in a two-variable potential V ( P, S ) and those of 3 bodies with arbitrarymasses and potential m + m + m m m m V ( P m , S m ) do coincide! The Hamiltonian (80) describes a two-dimensional particle moving in curved space or, equiv-alently, two-dimensional solid body in external potential V . The trajectories of H m aresolutions to the equations of motion for H Q (78) as well.The corresponding metric for (80) takes the form g µνm = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m + m + m m m m P m m + m + m m m m S m m + m + m m m m S m m + m + m m m m P m S m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (81)30ith determinant ˜ D m = ( m + m + m ) m m m ) S m [ P m − S m ] , cf.(66). It is worth emphasizing that the metrics (81) and (66) coincide up to a multiplicativefactor. From (81) it follows that the corresponding Ricci scalar Rs is given by˜Rs = m m m P m ( S m − m + m + m ) S m [ P m − S m ] = ( m + m + m ) S m P m ( S m − m m m ) ˜ D m , (82)cf.(67).In the particular case of a one-variable potential, V = m + m + m m m m V m ( P m ) , as it follows from (80), the Hamiltonian H Q (78) admits trajectories depending on P m only.They lie on the intersection between the hypersurfaces P Q = 0 and P S m = 0, they aregoverned by the two-dimensional Hamiltonian˜ H m ≡ H m | P Sm =0 = m + m + m m m m (cid:20) P m P P m + V ( P m ) (cid:21) , (83)cf.(68).As concrete example one can consider the mass-dependent 3-body anharmonic oscillatorpotential V ( AHO ) m = A P m + B P m , cf.(70), where A, B are parameters. As mentioned above, it is evident that multiplying theHamiltonian ˜ H m (83) by m m m m + m + m we arrive at the same Hamiltonian H P (71). Hence,the trajectories of two 3-body anharmonic oscillators defined by (71) and (83), respectively,coincide and, in general, they are given by (74).In the case of particular 3-body anisotropic harmonic oscillator with different masseswhich occurs at B = 0, see above, V ( HO ) m = A P m = A (cid:18) r m + r m + r m (cid:19) , the spring constants in (36) take values ν = ω m , ν = ω m and ν = ω m . It leadsto the periodic trigonometric solution P m ( t ) = c cos ( √ A t + c ) , where c > c are constants of integration.31 I. CONCLUSIONS
In this paper, which is the first in a series, the 3-body classical system in a d -dimensionalcoordinate space R d , d ≥
2, is considered. The study is restricted to potentials V = V ( r , r , r ) that depend solely on the relative distances between bodies. Assumingzero total angular momentum in the center-of-mass frame, the original 3 d − dimensionalproblem in configuration space R d is reduced to a 3 − dimensional space (of relative mo-tion) parametrized by mutual distances. The corresponding six-dimensional Hamiltonianis constructed explicitly in the space of relative motion. A new observation is that in the r − representation the Hamiltonian H r (18) at zero potential (the free problem) possessesformally a Z ⊗ -symmetry r ij → − r ij . This allows us to code this symmetry by introducingthe ρ ij = r ij variables and to proceed to the ρ − representation obtaining the Hamiltonian H (27). This Hamiltonian corresponds to the kinetic energy of 3-dimensional solid body.Making the identification of the coefficients of the tensor of inertia in H as the entriesof a contravariant metric (cometric), the emerging Hamiltonian describes a 3-dimensionalclassical particle moving in a curved space with cometric g µ ν ( ρ ) (23). Its kinetic energy isa polynomial in coordinate and momentum variables and, unlike the quantum case , is notaccompanied by effective potential. The determinant of cometric Det (cid:2) g µ ν ( ρ ) (cid:3) is of definitesign. It vanishes either at the triple-body collision point or on collinear configurations only.This cometric does not become singular at the two-body collision points unlike as it is inthe r − representation, already noticed in Ref. and Ref. for d = 2 and d = 3, respectively,as un undesirable property. The free problem in both r − and ρ − representations is S -permutationally invariant wrt the interchange of body positions and their masses.The positions of the 3 bodies form a triangle. In the case of three equal masses m = m = m = 1, the kinetic energy possesses an accidental S -symmetry wrt theinterchange of the edges (intervals of interaction) of this triangle. The full Z ⊕ S symme-try of the free problem is encoded in the set of generalized coordinates P , S and T obtainingthe Hamiltonian H geo (41) in the geometrical-representation . The two coordinates P and S are called volume variables . The variable P corresponds to the sum of squares of sides ofthe triangle whilst S is the area squared. In ( P, S, T )-variables, the kinetic energy remainspolynomial. It also describes a three-dimensional particle moving in a curved space with a32 -independent metric g µ ν ( P, S, T ). The determinant Det (cid:2) g µ ν ( P, S, T ) (cid:3) is again of definitesign. It vanishes when the triangle of interaction is isosceles. The properties and simplicity ofgeometrical-representation were illustrated in (I) the physically relevant 3-body Newtoniangravity potential ( d = 3) where (55) describes all four possible 3-body Coulomb/Newtonpotentials in a unified manner, and (II) in the planar 3-body choreographic motion onalgebraic lemniscate ( d = 2) for which the two geometrical variables P and T become particular constants of motion and the dynamics is parametrized by a planar elliptic curve S = ℘ ( t ; P, T ).The volume variables admit a generalization, S m (76) and P m (77), to systems witharbitrary masses. For potentials that solely depend on these variables, V = V ( P m , S m ),the trajectories in the volume-representation are governed by a four-dimensional Hamilto-nian H vol (63). It describes a two-dimensional particle moving in a curved space. In thevolume-representation, the trajectories are mass-independent. In the particular case of aone-variable potential, V = V ( P m ), the system is described by a two-dimensional Hamil-tonian only. In this representation, a shifted harmonic potential (equivalently, a 3-bodymass-dependent anharmonic potential in r -variables) was analyzed in detail. The volume-representation implies an effective reduction of the problem beyond separation of variables.Finally, in the general n -body case with n ≤ d − polytope of interaction . The volume-representation can be constructedimmediately. At fixed n there exist ( n −
1) volume-variables made out of elements withdifferent dimensionality (edges, faces, cells and so on) of the polytope of interaction. Itreveals a surprising link between the theory of regular polytopes and the dynamics of an n -body system. This will be presented in a forthcoming paper. VII. ACKNOWLEDGMENTS
A.M. thanks T. Fujiwara and E. Pi˜na for important remarks and personal discussions.R.L. is supported in part by CONACyT grant 237351 (Mexico). A.V.T. thanks Don Saari fora clarification, he is supported in part by the PAPIIT grant
IN113819 (Mexico). W.M. ispartially supported by a grant from the Simons Foundation (
EFERENCES Murnaghan F D, 1936
A Symmetric Reduction of the Planar Three-Body ProblemAm. J. Math. Van Kampen E R and Wintner Aurel, 1937
On a Symmetrical Canonical Reduction of theProblem of Three BodiesAm. J. Math. Turbiner A V, Miller W, Jr, and Escobar-Ruiz M A, 2018
Three-body problem in d -dimensional space: ground state, (quasi)-exact-solvabilityJ. Math. Phys. D M Y Sommerville, 1958
An Introduction to the Geometry of n Dimensions , New York:Dover, p.124 Fujiwara T, Fukuda H and Ozaki H, 2003
Choreographic Three Bodies on the LemniscateJ. Phys. A: Math. Gen. Turbiner A V and Lopez Vieyra J C, 2019
Particular superintegrability of 3 body NewtonianGravity arXiv:1910.11644v1 Turbiner A V, 2013
Particular Integrability and (Quasi)-exact-solvability J. Phys. A: Math.Theor. Landau L D, Lifshitz, 1976
Mechanics
Vol. 1 §
41 Pergamon Press Turbiner A V, Miller W, Jr, and Escobar-Ruiz M A 2020
Three-body closed chain ofinteractive (an)harmonic oscillators and the algebra sl (4 , R ) J. Phys. A: Math. Theor. Lagrange J L, 1772
Essai sur le problme des trois corps , Oeuvres, Vol. 6. Moeckel R, 2014
Central configurations , Scholarpedia, 9(4):10667. Saari D G, 1980
On the role and the properties of central configurations in the n-bodyproblem Celestial Mechanics Albouy A and Chenciner A, 1998
Le probl`eme des n corps et les distances mutuelles Inv.Math. Moore C, 1993
Braids in classical dynamics Phys Rev Lett Chenciner A and Montgomery R, 2000
A remarkable periodic solution of the three-bodyproblem in the case of equal masses Ann. Math. Lopez Vieyra J C, 2019
Five-body choreography on the algebraic lemniscate is a potentialmotion Physics Letters A Wiggins S, 1990
Invariant Manifolds: Linear and Nonlinear Systems §§