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Arnold Diffusion in the Full Three-Body Problem

Maciej J. Capinski, Marian Gidea

TL;DR

This work establishes Arnold diffusion in the planar full three-body problem by casting the TBP as a perturbation of the Kepler problem and the PCR3BP and proving energy transfer between these components via a computer-assisted, topological method based on correctly aligned windows. It provides a concrete, physically relevant instance—the Neptune–Triton–asteroid system—where diffusion occurs for asteroid masses up to a calculable bound, with explicit diffusion times. The approach relies on reducing the problem to section-to-section maps along a homoclinic orbit to a Lyapunov orbit, introducing energy-consistent coordinates that link the Kepler energy $\bar{K}_{\varepsilon}$ and the PCR3BP energy $\bar{H}_{\varepsilon}$, and validating a finite family of connecting sequences via interval arithmetic (CAPD). The results demonstrate macroscopic, parameter-robust instability in a realistic celestial system and illustrate the power of topological, computer-assisted proofs for high-dimensional Hamiltonian dynamics.

Abstract

The full three-body problem, on the motion of three celestial bodies under their mutual gravitational attraction, is one of the oldest unsolved problems in classical mechanics. The main difficulty comes from the presence of unstable and chaotic motions, which make long-term prediction impossible. In this paper, we show that the full three-body problem exhibits a strong form of instability known as Arnold diffusion. We consider the planar full three-body problem, formulated as a perturbation of both the Kepler problem and the planar circular restricted three-body problem. We show that the system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- of an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We demonstrate that the approach can be applied to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the diffusion time.

Arnold Diffusion in the Full Three-Body Problem

TL;DR

This work establishes Arnold diffusion in the planar full three-body problem by casting the TBP as a perturbation of the Kepler problem and the PCR3BP and proving energy transfer between these components via a computer-assisted, topological method based on correctly aligned windows. It provides a concrete, physically relevant instance—the Neptune–Triton–asteroid system—where diffusion occurs for asteroid masses up to a calculable bound, with explicit diffusion times. The approach relies on reducing the problem to section-to-section maps along a homoclinic orbit to a Lyapunov orbit, introducing energy-consistent coordinates that link the Kepler energy and the PCR3BP energy , and validating a finite family of connecting sequences via interval arithmetic (CAPD). The results demonstrate macroscopic, parameter-robust instability in a realistic celestial system and illustrate the power of topological, computer-assisted proofs for high-dimensional Hamiltonian dynamics.

Abstract

The full three-body problem, on the motion of three celestial bodies under their mutual gravitational attraction, is one of the oldest unsolved problems in classical mechanics. The main difficulty comes from the presence of unstable and chaotic motions, which make long-term prediction impossible. In this paper, we show that the full three-body problem exhibits a strong form of instability known as Arnold diffusion. We consider the planar full three-body problem, formulated as a perturbation of both the Kepler problem and the planar circular restricted three-body problem. We show that the system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- of an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We demonstrate that the approach can be applied to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the diffusion time.

Paper Structure

This paper contains 18 sections, 6 theorems, 143 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries -- the three-body problem
  3. The full three-body problem as a perturbation of the Kepler problem and the circular restricted three-body problem
  4. The Hamiltonian of the three--body problem in rotating Jacobi coordinates
  5. The three-body problem as a perturbation of the Kepler and the circular restricted three-body problem
  6. Geometry of the restricted three-body problem
  7. Statement of the main result
  8. Surfaces of sections and local coordinates
  9. Choices of sections
  10. Choices of local coordinates on sections
  11. Tools for establishing diffusion
  12. Correctly aligned windows and cone conditions
  13. Diffusion mechanism
  14. Proof of the main result
  15. The numerical properties of the homoclinic orbit
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the planar full three-body problem, where the masses of the bodies are $m_0,m_1$ and $m_2=\varepsilon^2$. For fixed values of $m_0$ and $m_1$, as specified below, and for a mass parameter $\varepsilon ^{2}$, from arbitrarily close to $0$ up to some $\varepsilon_0^2$, there exist trajectorie

Figures (7)

  • Figure 1: Jacobi coordinates for the three body-problem.
  • Figure 2: On the right hand side we have a homoclinic orbit to one of the Lyapunov orbits of the PCR3BP. The points $q_i$, for $i=0,\ldots,122$, are positioned along the homoclinic in the $(\bar{r}_2,\bar{\phi}_2,\bar{R}_2,\bar{\Phi}_2)$ coordinates. For all the points we choose $\bar{r}_1=0$ and $\bar{R}_1=\frac{1}{10}$.
  • Figure 3: An example of correctly aligned windows $N\overset{f}{\implies }M$. The exit sets $N^-$ and $M^-$ are depicted in grey. The entry sets $N^+$ and $M^+$ consist of the white sides of the cubes $N$ and $M$, respectively. Here $cs=(s,\alpha,I)$, where $s$ is contracting and $\alpha,I$ are center coordinates.
  • Figure 4: Cone condition. A gray cone at $z$ is mapped inside a white cone at $f(z)$.
  • Figure 5: The homoclinic orbit from Figure \ref{['fig:surfaces']} plotted in $\alpha ,\bar{r}_{2}$ coordinates.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Theorem 1: Main theorem
  • Lemma 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Lemma 9
  • Remark 10
  • ...and 16 more