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Chaotic dynamics for the two body problem on a sphere

Sergey Bolotin

TL;DR

This work proves real-analytic nonintegrability and chaotic dynamics for the two-body problem on the sphere $S^2$ in the regime of a small lighter body mass. It develops and applies a general anti-integrable-limit framework for Lagrangian systems with Newtonian singularities, reducing the full problem to a restricted model that captures near-collision behavior. The authors show the existence of a hyperbolic invariant set on relevant energy and angular-momentum levels, conjugate to a subshift of finite type, and then extend the result to the full two-body problem by perturbation arguments in the mass ratio. The results illuminate chaotic trajectories akin to Poincaré’s second species and establish analytic nonintegrability in a curved-space celestial-mechanics setting with near-collision dynamics.

Abstract

We prove the existence of chaotic trajectories for the two body problem on a sphere. The trajectories we construct encounter near-collisions and are similar to the second species solutions of Poincaré of the classical 3 body problem. The construction uses a general result on Lagrangian systems with Newtonian singularities of the potential which is based on the method of anti-integrable limit of Serge Aubry.

Chaotic dynamics for the two body problem on a sphere

TL;DR

This work proves real-analytic nonintegrability and chaotic dynamics for the two-body problem on the sphere in the regime of a small lighter body mass. It develops and applies a general anti-integrable-limit framework for Lagrangian systems with Newtonian singularities, reducing the full problem to a restricted model that captures near-collision behavior. The authors show the existence of a hyperbolic invariant set on relevant energy and angular-momentum levels, conjugate to a subshift of finite type, and then extend the result to the full two-body problem by perturbation arguments in the mass ratio. The results illuminate chaotic trajectories akin to Poincaré’s second species and establish analytic nonintegrability in a curved-space celestial-mechanics setting with near-collision dynamics.

Abstract

We prove the existence of chaotic trajectories for the two body problem on a sphere. The trajectories we construct encounter near-collisions and are similar to the second species solutions of Poincaré of the classical 3 body problem. The construction uses a general result on Lagrangian systems with Newtonian singularities of the potential which is based on the method of anti-integrable limit of Serge Aubry.

Paper Structure

This paper contains 6 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Restricted 2 body problem
  4. Lagrangian systems with singularities
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Theorem \ref{['thm:2body']}

Key Result

Theorem 1

Fix constants $-1/2<a<b$ and let $\varepsilon_0>0$ and $\sigma_0>0$ be sufficiently small. If $m_2/m_1<\sigma_0$, the 2-body problem on $S^2$ has a chaotic invariant set on the level sets $\{H=h,\mathbf{M}=\mathbf{M}_0\}$ of the first integrals for

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4