AAN INTRODUCTION TO THE KEPLER-HEISENBERG PROBLEM
COREY SHANBROM Introduction
Here we provide an overview of what is known, and what is not known, about an interestingdynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a versionof the classical Kepler problem of planetary motion, but in a sub-Riemannian setting. Theresult is system which is surprisingly rich and beautiful, mysterious in some ways but tame inothers, offering a substantial number of questions which seem non-trivial yet tractable.1.1.
History and motivation.
The curved Kepler problem, typically set in a Riemannianmanifold of constant curvature, is an active area with a fascinating history. In particular, whilethe co-discovery of hyperbolic space by Lobachevsky and Bolyai is famous mathematical lore,both men also independently attempted to pose the Kepler problem in this new geometry around1835. Dirichlet and Schering worked on this problem shortly after. In 1873, Lipschitz posedand partially solved the Kepler problem on the 3-sphere, as did Killing (with a different notionof gravity) in 1885. In 1902, Liebmann proved that orbits are conics in both hyperbolic andand spherical 3-space. There is a parallel history of studying the Kepler problem on surfaces aswell, and today there is active work on n -body problems in more general curved spaces. Detailscan be found in [1] and the references therein (while this paper is published, the arXiv versioncontains a much more elaborate history).The original motivation for studying the Kepler-Heisenberg problem was to extend this ideato sub-Riemannian manifolds, which generalize Riemannian ones. In particular, the three di-mensional Heisenberg group is the simplest non-trivial sub-Riemannian space and the naturalplace to begin such an investigation. While no knowledge of sub-Riemannian geometry is neededfor this article, the interested reader can see [9] for an introduction to this subject in general, anddetails of the Heisenberg group in particular. As seen below, the resulting Kepler-Heisenbergsystem retains certain aspects of the classical Kepler problem in Euclidean space, while reflect-ing the unique geometry of the Heisenberg group. We are thus left with obvious questions aboutgeneralizing both the setting to other sub-Riemannian spaces and the force to other classicalmechanical systems. See Section 3.We can provide an alternative motivation as well. The classical Kepler problem enjoys thefamous three Kepler’s Laws of Planetary Motion. Do these hold in curved spaces? For sphericaland hyperbolic 3-space, the first two laws hold but the third fails. Why? Kepler’s third law is aconsequence of homogeneity of the gravitational potential with respect to Euclidean dilations.But spherical and hyperbolic spaces do not admit such dilations. Moreover, by a result ofGromov ([7]), there are no homogeneous Riemannian manifolds which admit dilations besidesEuclidean spaces. Thus, if one wants to pose a curved Kepler problem in which Kepler’s thirdlaw holds, one must leave the Riemannian realm. Certain important sub-Riemannian spacesknown as Carnot groups do admit dilations. The Heisenberg group is the simplest, and we willsee below that a version of Kepler’s third law does indeed hold there.1.2. The problem.
We treat the Heisenberg group H as the smooth manifold R equippedwith a certain non-Euclidean metric. Take the usual global coordinates ( x, y, z ) and define twovector fields X = ∂∂x − y ∂∂z and Y = ∂∂y + 12 x ∂∂z . Date : January 12, 2021. a r X i v : . [ m a t h . D S ] J a n hese span a two-dimensional bracket-generating distribution D . The Kepler-Heisenberg prob-lem is roughly the classical Kepler problem in R with the constraint that all orbits are tangentto D ; such curves are called horizontal . Declaring that X and Y are orthonormal defines aninner product on D . Then H is now equipped with a sub-Riemannian structure whereby we canmeasure lengths of horizontal curves and (by the Chow-Rashevskii theorem) distances betweenpoints. The resulting geometry enjoys three interesting properties: • the length of a horizontal curve is equal to the Euclidean length of its projection to the xy -plane ( ds = ( dx + dy ) | D ), • the z -coordinate of a horizontal curve grows like the area traced out by its projection(Stokes’ theorem), • geodesics are helices projecting to circles or lines (dual isoperimetric problem).Our configuration space is H equipped with this structure. Here we pose the Kepler-Heisenberg problem as a Hamiltonian system, taking T ∗ H with coordinates ( x, y, z, p x , p y , p z )as phase space. Consider the dual momenta to the vector fields X and Y given by P X = p x − yp z and P Y = p y + xp z . Then, as in the Riemannian case, we take our kinetic energy to be K = ( P X + P Y ). Thisis induced by our metric and generates geodesic flow. Choosing the gravitational potentialis more subtle. In the classical case, the potential 1 /r has many important properties. Itis most obviously the reciprocal of distance from the origin, but for most (sub-)Riemannianmanifolds there is no explicit expression for such distance, so this characterization is inadequate.Instead, we recognize 1 /r as (a multiple of) the fundamental solution to the Laplacian on R , and choose our potential accordingly. Thus we choose the Kepler-Heisenberg potential as U = − π (cid:16) ( x + y ) + 16 z (cid:17) − / , which is the fundamental solution to the Heisenberg sub-Laplacian ∆ H = X + Y according to [6]. Note the singularity at the origin, which representsour planet colliding with the sun.Thus we obtain our Hamiltonian H = (( p x − yp z ) + ( p y + xp z ) ) (cid:124) (cid:123)(cid:122) (cid:125) K − π (cid:112) ( x + y ) + 16 z (cid:124) (cid:123)(cid:122) (cid:125) U . The
Kepler-Heisenberg problem is to analyze and eventually solve Hamilton’s equations for H .2. Results
We first give some basic properties of the system, most of which can be found in [10]. Animportant tool will be the Heisenberg dilation ( x, y, z ) (cid:55)→ ( λx, λy, λ z ) for λ > Proposition 1.
The Kepler-Heisenberg system has two integrals of motion: the total energy H and the angular momentum p θ = xp y − yp x . The dilational momentum J = xp x + yp y + 2 zp z satisfies ˙ J = 2 H . The dynamics are therefore integrable on the codimension one submanifold { H = 0 } , where J is also conserved. Periodic orbits must satisfy H = J = 0 . These are the known symmetries – there may be others. So we do not know whether thissystem is integrable or not. But we consider it ‘at least mostly integrable’.Ostensibly, we should be able to integrate the zero-energy dynamics. However, the dilationalaction is non-compact, so the standard machinery of action-angle variables does not immediatelyapply. (While there is a generalized theory, as in [5], it is non-constructive.) Our invariantsubmanifolds can be visualized as cones of 2-tori, T × R + , with the cone point at 0 ∈ R + representing the singularity at sun. If an orbit’s constant dilational momentum is positive(resp. negative) then it winds away (resp. towards) the cone point; if J = 0 then the orbitstays on a compact T and the motion is quasi-periodic or periodic. In [10], the integration ofthe H = 0 system is reduced to parametrizing a family of degree 6 algebraic curves, but thereis no general method for obtaining such a parametrization. igure 1. Periodic orbits with rotation numbers 5/9, 6/41, 1/4, and 7/8. Inall figures we show the projection of the orbit to the xy -plane.; the z -coordinategrows like the area traced out.As mentioned in the Introduction, we also have a version of Kepler’s third law: T = ka where T represents a orbit’s period, a is its size, and k is a universal constant. This is a con-sequence of our Hamiltonian being homogeneous of degree − xy -plane are lines through the origin.On the invariant submanifold { z = p z = p θ = 0 } we have a classical central force problem in theplane, and the radial distance r ( t ) traces out a hyperbola, ellipse, or parabola if the energy ispositive, negative, or zero, respectively. Lastly, there is one very strange family of orbits: eachsits at a constant position on the z -axis while its momenta p z grows linearly in time.2.1. Periodic orbits.
The existence of periodic orbits was established in [11]. However, theproof followed the direct method in the calculus of variations and provided very little informationabout such orbits beyond their existence. Numerical methods in [3] allowed the discovery of avery rich and beautiful symmetry structure.
Theorem 2.
For any rational j/k ∈ (0 , , there is a periodic Kepler-Heisenberg orbit withrational rotation number j/k . Rational rotation numbers are treated in [8, 12]; here it simply means that after one k th ofthe period, the planet has rotated 2 π · jk radians counterclockwise about the z -axis. See Figures1 and 2. All such orbits possess k -fold rotational symmetry about the z -axis, except the orbitwith rotation number 1/1 (its projection does, but not z ( t )).This symmetry structure was uncovered via numerical explorations. We strongly encour-age other researchers to use our codebase [2]. All code is hosted publicly as free, open-sourcesoftware; explicit instructions to reproduce each plot are provided there, with all relevant doc-umentation and licensing. All results in [3] are thus 100% available and exactly reproducible. ymmetry type 1:1 Symmetry type 1:2Symmetry type 1:3 Symmetry type 2:3 Symmetry type 1:4 Symmetry type 3:4Symmetry type 1:5 Symmetry type 2:5 Symmetry type 3:5 Symmetry type 4:5 Figure 2.
All periodic orbits with k ≤ X is close enough to a closed orbit. To do so we use a symplectic integrator andshooting method with objective function obj giving the smallest local minimum of distance from X . If X is satisfactory, we use a Monte Carlo method to optimize obj as follows. Randomlypick a point in radially symmetric distribution on an annular neighborhood centered at X ,then symplectically integrate and evaluate obj . If obj does not decrease, try a new point; if obj decreases, use this point as new X . Repeat this update step 1000 times. The output is animage, a binary data file, and a human readable summary containing all pertinent informationabout the orbit (initial conditions, angular momentum, period, etc.). The program also detectsthe rotation number using Fourier analysis.Finally, our numerical investigations discovered another somewhat surprising phenomenonthat we do not fully understand. Due to the symmetries and constraints of the system, we canreduce the search for periodic orbits to one dimension, which we can parametrize by angularmomentum p θ . Then as a function of p θ , the rotation numbers are arranged according to theFarey sequence. See Table 1. Table 1. As p θ grows, the rotation numbers are distributed like the Farey sequence. ≈ p θ jk
11 56 45 34 23 35 12 25 13 14 15 164 .2.
Self-similarity.
Zero-energy orbits are relatively well behaved and understood. Almostall exhibit an attractive self-similarity property ([4]). To the best of our knowledge thesesolutions represent the first occurrence of this type of self-similarity within solutions to ODEsof Hamiltonian type.
Theorem 3.
Suppose c is a curve in T ∗ H such that: • c solves Hamilton’s equations • H ( c ( t )) = 0 • z has consecutive zeros at t , t , t Then c is entirely determined by its value on the fundamental domain [ t , t ] . See Figure 4, which is explained in detail below. The self-similarity here involves a rotation,a dilation, and a time reparametrization. The first two assumptions in the theorem are clear,but the third requires some explanation. Numerical evidence suggests the following.
Conjecture 4.
All zero-energy orbits avoiding the z -axis have z ( t ) oscillatory. By oscillatory we mean that z ( t ) has infinitely many zeros without being identically zero onany interval, which of course implies the existence of three consecutive zeros needed for ourtheorem. We tried and failed to prove this conjecture using Sturm comparison methods. It isnecessary that we avoid the z -axis in this conjecture; we have counterexamples otherwise. Butthe z -axis is special in the Heisenberg group – it is the cut and conjugate locus for the origin– and needs to be treated with care. However, numerical evidence suggests that even thosezero-energy orbits which do intersect the z -axis are self-similar, although perhaps in a differentsense than those which do not intersect the z -axis, with z ( t ) not necessarily oscillatory. Weprovide details at the end of this section. The upshot is the following. Conjecture 5.
All zero-energy orbits are self-similar.
The self-similarity in Theorem 3 is most clearly understood in new coordinates. Let s = 14 log(( x + y ) + 16 z ) θ = arg( x, y ) u = arg( x + y , z ) . Then the corresponding momenta p s and p θ correspond precisely to the dilational and angularmomenta; that is, p s = J . This allows us to compute the dilational and rotational factors λ and ϕ for any given orbit satisfying our hypotheses as λ = exp( s ( t ) − s ( t )) and ϕ = θ ( t ) − θ ( t ) . The rotational part is standard, but finding the correct coordinates for the dilational partrequires solving a system of non-linear PDEs. Armed with the dilational factor λ we can nowexplicitly determine the time reparametrization τ as follows (if λ = 1 then no reparametrizationis necessary): ξ ( t ) = 12 log λ (cid:18) − ( t − t ) 1 − λ t − t (cid:19) τ ( t ) = t + ( t − t ) 1 − λ ξ ( t )+2 (cid:98) ξ ( t ) (cid:99) − λ . Suppose we are given a segment of a curve c on a fundamental time domain [ t , t ], and wewant to know c ( t ) at some future time t . Then c can be extended from [ t , t ] to next fundamentaldomain [ t , t + λ ( t − t )] by appropriately rotating, dilating, and reparametrization time. Wecan then iterate to move forward to any future time (until possible collision), and can easilymodify to move backward to past time. The function ξ plays an important discretizing role:its floor gives the number of iterations needed. That is, (cid:98) ξ ( t ) (cid:99) is the number of rotated anddilated copies of c which need to be pasted together in order to reach c ( t ). Note that consecutive“periods” scale by λ , giving a geometric series, so we have the following corollary. able 2. Summary of behaviorDilational momentum Dilation factor Behavior of orbit
J < λ < J > λ > J = 0 λ = 1 Periodic or quasi-periodic motion Graph: (x(t), y(t))
Graph: (t, z(t))Trajectory with H = 0, J < 0
Graph: (x(t), y(t))
Graph: (t, z(t))Trajectory with H = 0, J > 0
Graph: (x(t), y(t))
Graph: (t, z(t))Trajectory with H = 0, J = 0
Figure 3.
Sample orbits with negative, positive, zero dilational momentum.
Corollary 6.
Orbits satisfying the hypotheses of Theorem 3 stratify according to Table 2. Allcollisions happen in finite time with t col = t + t − t − λ . Figure 3 provides sample orbits from the stratification in Table 2.Most of the self-similarity discussed in this section is apparent in Figure 4, which displaysan orbit with negative dilational momentum. On the left we see the projection of the orbitto the xy -plane, on the right we see z as a function of time. In both images the blue curverepresents the orbit itself (numerically integrated), the purple segment represents the part overa chosen fundamental domain, and the orange segment is a rotated and dilated version of thepurple. The idea is that we are given the purple segment, from which we can recover the orangesegment, which in turn matches the blue curve. This can be iterated forward or backward intime. The vertical lines on the right show zeros of z ( t ); the given fundamental domain [ t , t ]is shown with bold lines. We can see how these “periods” shrink geometrically and a collisionoccurs in finite future time. The codebase used to create Figure 4 will also be available at [2].To conclude this section we revisit Conjecture 5 and the discussion preceding it. Recall thatthe z -axis is the conjugate and cut locus for the origin in the Heisenberg group. Zero-energyorbits meeting the z -axis need not have z ( t ) oscillatory, but we believe they are all still self-similar. However, numerical investigations suggest that such orbits bifurcate into two familieswith qualitatively different behavior, and this bifurcation occurs when dilational momentum | J | = √ π . .5 0.0 0.5 1.00.50.00.51.01.5 0 5 10 15 20 25 300.20.10.00.10.20.30.40.5 Figure 4.
A self-similar orbit.
Conjecture 7.
Consider a zero-energy orbit meeting the z -axis. If | J | < √ π then we haveself-similarity like a generic H = 0 orbit. In particular, z ( t ) oscillates and the orbit looks likea dilating figure eight. If | J | > √ π then we have self-similarity like Heisenberg geodesic. Inparticular, z ( t ) is monotonic and the orbit looks like a dilating helix. See Figures 5 and 6. Figure 5.
Starting on z -axis with small dilational momentum. Figure 6.
Starting on z -axis with large dilational momentum . Open Questions
Here we simply list some of the many questions we have about this system and its general-izations. Some of these we have investigated without success, others we have barely considered.(1) Is the Kepler-Heisenberg system integrable? Are there any other non-trivial symmetries?(2) How can we explicitly integrate the zero-energy system?(3) How do orbits with nonzero energy behave? Nearly all our efforts have focused on the H = 0 case, where the dynamics are integrable and where the periodic orbits live.(4) Are Conjectures 4, 5, 7 true?(5) Can we regularize collisions with the sun?(6) Can we quantize this system in a meaningful way?(7) The richness of this system gives us hope that other “sub-Riemannian mechanics” prob-lems may be worth studying. In each case we would take the sub-Riemannian Hamil-tonian as our kinetic energy and choose some appropriate potential energy. But thisgives freedom to choose both the geometry and the potential:(a) Can we study the Kepler problem on higher dimensional Heisenberg groups? OtherCarnot groups? Other sub-Riemannian spaces? Is there a non-Euclidean settingwhere all three Kepler laws hold?(b) Can we study the two body problem (not same as Kepler!) on the Heisenberggroup? Three bodies? Other classical mechanics problems? Other potentials?(c) Combine the previous two items. The n -body problem in jet space? References [1] F. Diacu, E. Perez-Chavela, and M. Santoprete, The n -body problem in spaces of constant curvature,arXiv:0807.1747 (2008).[2] V. Dods, “Kepler-Heisenberg Problem Computational Tool Suite.” Available at https://github.com/vdods/heisenberg , 2018.[3] V. Dods and C. Shanbrom, Numerical methods and closed orbits in the Kepler-Heisenberg problem, Exp.Math. , (2019), 420–427.[4] V. Dods and C. Shanbrom, Self-similarity in the Kepler-Heisenberg problem, to appear in J. Nonlinear Sci. ,arXiv:1912.12375.[5] E. Fiorani, G. Giachetta, and G. Sardanashvily, The Liouville-Arnold-Nekhoroshev theorem for non-compactinvariant manifolds,
J. Phys. A , (2003), L101-–L107.[6] G. Folland, A fundamental solution to a subelliptic operator, Bull. Amer. Math. Soc. , (1973), 373–376.[7] M. Gromov, Metric structures for Riemannian and Non-Riemannian Spaces , Birkhauser (2007).[8] A. Katok and B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems , Cambridge Univer-sity Press (1995).[9] R. Montgomery,
A Tour of Subriemannian Geometries, Their Geodesics and Applications , AMS Mathemat-ical Surveys and Monographs, (2002).[10] R. Montgomery and C. Shanbrom, Keplerian motion on the Heisenberg group and elsewhere, Geometry,mechanics, and dynamics: The Legacy of Jerry Marsden , Fields Inst. Comm., (2015), 319–342.[11] C. Shanbrom, Periodic orbits in the Kepler-Heisenberg problem, J. Geom. Mech. , (2014), 261–278.[12] S. Tabachnikov, Geometry and Billiards, AMS Student Mathematical Library, (2005). California State University, Sacramento, 6000 J St., Sacramento, CA 95819, USA
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