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Consecutive collision orbits in the restricted three-body problem above the first critical energy value

Jungsoo Kang, Kevin Ruck

TL;DR

This work addresses the existence of symmetric collision-type orbits in the planar circular restricted three-body problem for energy levels just above the first critical value. By regularizing collisions via the Birkhoff and Moser frameworks and leveraging a $\mathbb{Z}_2$-equivariant Rabinowitz Floer homology on the model $S^1\times S^2$, the authors connect Reeb chords to symmetric consecutive collisions. They prove that the energy hypersurface is of contact type and derive a dichotomy: either a periodic symmetric collision orbit exists or infinitely many symmetric consecutive collision orbits occur, with a genericity argument ruling out odd-numbered collisions for periodic cases, ensuring at least two distinct symmetric consecutive orbits. The results integrate regularization geometry, Reeb dynamics, and equivariant Floer techniques to advance a variational framework for orbit existence above the first energy threshold in celestial mechanics.

Abstract

In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy levels are of contact type. Then, using a version of Rabinowitz Floer homology, we establish the existence of either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits. Furthermore, by an analytic continuation argument, for generic mass ratios and energy levels, we prove that there is no periodic symmetric collision orbit with odd number of collisions. This in turn implies the existence of at least two symmetric consecutive collision orbits.

Consecutive collision orbits in the restricted three-body problem above the first critical energy value

TL;DR

This work addresses the existence of symmetric collision-type orbits in the planar circular restricted three-body problem for energy levels just above the first critical value. By regularizing collisions via the Birkhoff and Moser frameworks and leveraging a -equivariant Rabinowitz Floer homology on the model , the authors connect Reeb chords to symmetric consecutive collisions. They prove that the energy hypersurface is of contact type and derive a dichotomy: either a periodic symmetric collision orbit exists or infinitely many symmetric consecutive collision orbits occur, with a genericity argument ruling out odd-numbered collisions for periodic cases, ensuring at least two distinct symmetric consecutive orbits. The results integrate regularization geometry, Reeb dynamics, and equivariant Floer techniques to advance a variational framework for orbit existence above the first energy threshold in celestial mechanics.

Abstract

In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy levels are of contact type. Then, using a version of Rabinowitz Floer homology, we establish the existence of either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits. Furthermore, by an analytic continuation argument, for generic mass ratios and energy levels, we prove that there is no periodic symmetric collision orbit with odd number of collisions. This in turn implies the existence of at least two symmetric consecutive collision orbits.

Paper Structure

This paper contains 8 sections, 5 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regularization of the Restricted Three-Body Problem
  3. Moser regularization
  4. Birkhoff Regularization
  5. Contact property of Birkhoff regularized energy hypersurfaces
  6. Reeb dynamics on $S^1\times S^2$
  7. Contactomorphism to the standard contact structure
  8. Indices of Reeb chords on $S^1\times S^2$

Key Result

Theorem 1

( albers2012a) For every $c<H(L_1)$, the Moser-regularized hypersurface $\overline{\Sigma}_c^{\mathfrak{M},E}$ is transverse to the standard Liouville vector field on $T^*S^2$. In particular, the restriction of the Liouville one-form on $T^*S^2$ to $\overline{\Sigma}_c^{\mathfrak{M},E}$ is a contac

Figures (2)

  • Figure 1: Sketch of two (symmetric) consecutive collision orbits in the R3BP.
  • Figure 2: Interpretation of the above define Legendrian submanifolds in the position space picture of the unregularized setting.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Proposition 1
  • Lemma 1
  • Proof 1
  • Remark 2
  • Proposition 2
  • Proof 2