Nonintegrability of the two-body problem in constant curvature spaces

Shchepetilov, A. V.

doi:10.1088/0305-4470/39/20/011

Cited by 41 publications

(40 citation statements)

References 24 publications

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“…Several members of the Russian school of celestial mechanics, including Valeri Kozlov and Alexander Harin, [43], [45], Alexey Borisov, Ivan Mamaev, and Alexander Kilin, [5], [6], [7], [8], [39], Alexey Shchepetilov, [67], [68], [69], and Tatiana Vozmischeva, [74], revisited the idea of the cotangent potential for the 2-body problem and considered related problems in spaces of constant curvature starting with the 1990s. The main reason for which Kozlov and Harin supported this approach was mathematical.…”

Section: Introductionmentioning

confidence: 99%

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

2012

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Abstract. We generalize the Newtonian n-body problem to spaces of curvature κ = constant, and study the motion in the 2-dimensional case. For κ > 0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for κ < 0. In the end, we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. We also emphasize that fixed points are specific to the case κ > 0, hyperbolic relative equilibria to κ < 0, and Lagrangian orbits of arbitrary masses to κ = 0-results that provide new criteria towards understanding the large-scale geometry of the physical space.

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Section: Introductionmentioning

confidence: 99%

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

2012

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“…This is the case, for instance, of the two centre problem, the harmonic oscillator, and the n vortex problem. See [3,7,8,26,37] and the references therein.Contrary to the euclidean case, the two-body problem in H 2 does not reduce to the Kepler problem and is nonintegrable [34]. A number of publications have addressed the dynamics of the restricted two-body problem in H 2 (see [4,17] and the references therein).…”

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confidence: 99%

Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2

García-Naranjo

Marrero

Pérez‐Chavela

et al. 2016

Journal of Differential Equations

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“…In particular, it would be interesting to see what the flow on the full reduced space limits to. Furthermore, given that we have the Poisson structure on the full reduced space, it would be nice to see if this offers any use in demonstrating the non-integrability for the 2-body problem (see [Shc06]) or whether additional integrable systems can be found for different potentials. One might hope that this would connect with the substantial literature that exists for integrable systems on SO(4).…”

Section: Concluding Comments and The Scope For Further Workmentioning

confidence: 99%

Singular Reduction of the 2-Body Problem on the 3-Sphere and the 4-Dimensional Spinning Top

Arathoon

2019

Regul. Chaot. Dyn.

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We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case for when the sphere is 3-dimensional and where we take the group of symmetries to be SO(4). As the 3-sphere is a group, both left and right multiplication on itself are commuting symmetries which together generate the full symmetry group. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group SE(4). The full reduced spaces are generically 4-dimensional and we describe these spaces and their singular strata.The dynamics of the 2-body problem descend through a double cover to give a dynamical system on SO(4), which after reduction is the same as that of a 4-dimensional spinning top with symmetry. This connection allows us to 'hit two birds with one stone' and derive results about both the spinning top and the 2-body problem simultaneously. We provide the equations of motion on the reduced spaces and fully classify the relative equilibria and discuss their stability. arXiv:1904.00801v1 [math.DS] 1 Apr 2019By drawing an analogy with the reduction of the Lagrange top, where reduction is also done in stages by first reducing in the body frame and then in the space frame about the axes of symmetry, we invoke the Semidirect Product Reduction by Stages Theorem to express the left and right reduced spaces as coadjoint orbits of SE(4). This is entirely analogous to the situation for the Lagrange top, whose intermediate reduced spaces are coadjoint orbits of SE(3). As the actions of left and right translation are free, these reduced spaces are well-behaved smooth manifolds. However, to complete the full reduction by the residual left or right action we necessarily have to handle non-free and singular points of the momentum. We employ the methods of singular and universal reduction through the use of some invariant theory to describe these reduced spaces, which are generically 4-dimensional. We give the corresponding equations of motion on the full reduced space for both the 2-body problem and the spinning top, and explicitly exhibit an additional integral for the symmetric spinning top demonstrating complete integrability.We then turn our attention to the relative equilibria. Instead of classifying these by finding fixed points in the reduced space directly, we instead find solutions in the intermediate left and right reduced spaces which are the orbits of one-parameter subgroups. This pleasantly turns out to be comparatively easy, amounting to an entirely linear problem in Euclidean geometry. Having classified the solutions in the left reduced space, it is then only a matter of reconstruction to obtain the full classification of relative equilibria on the original space. We then explore the stability of the corresponding fixed points in the full reduc...

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Nonintegrability of the two-body problem in constant curvature spaces

Abstract: We consider the reduced two-body problem with a central potential on the sphere S 2 and the hyperbolic plane H 2 . For two potentials different from the Newton and the oscillator ones we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.

Cited by 41 publications

References 24 publications

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2

Singular Reduction of the 2-Body Problem on the 3-Sphere and the 4-Dimensional Spinning Top

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