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The two-boost problem and Lagrangian Rabinowitz Floer homology

Kai Cieliebak, Urs Frauenfelder, Eva Miranda, Jagna Wiśniewska

TL;DR

The paper introduces a Lagrangian Rabinowitz Floer homology (LRFH) framework to study the two-boost problem in a class of magnetic-like Hamiltonian systems on noncompact energy hypersurfaces, with a focus on planar circular restricted three-body type dynamics. It constructs the LRFH on cotangent bundles, establishes Morse-type nondegeneracy, action filtrations, and invariant continuation under compact perturbations, and proves key bounds for Floer trajectories using an array of analytic tools including the maximum principle. Applying this to the Copernican Hamiltonian, it shows the positive LRHF is well-defined and invariant, and for large energy there is exactly one positive-action Reeb chord whose Maslov index vanishes, yielding a single generator in degree zero and thus solvability of the two-boost problem in this regime. The work further discusses the extension to noncompactly supported potentials decaying sufficiently fast and outlines how the critical set structure persists under suitable perturbations, setting the stage for broader applications and future work on the planar RC3BP case with slower decay. Overall, the paper provides a rigorous variational and homological toolkit to certify reachability via two boosts in noncompact celestial mechanics models and identifies precise topological invariants governing such connections.

Abstract

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems related to the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The main technical work goes into dealing with the noncompactness of the corresponding energy hypersurfaces.

The two-boost problem and Lagrangian Rabinowitz Floer homology

TL;DR

The paper introduces a Lagrangian Rabinowitz Floer homology (LRFH) framework to study the two-boost problem in a class of magnetic-like Hamiltonian systems on noncompact energy hypersurfaces, with a focus on planar circular restricted three-body type dynamics. It constructs the LRFH on cotangent bundles, establishes Morse-type nondegeneracy, action filtrations, and invariant continuation under compact perturbations, and proves key bounds for Floer trajectories using an array of analytic tools including the maximum principle. Applying this to the Copernican Hamiltonian, it shows the positive LRHF is well-defined and invariant, and for large energy there is exactly one positive-action Reeb chord whose Maslov index vanishes, yielding a single generator in degree zero and thus solvability of the two-boost problem in this regime. The work further discusses the extension to noncompactly supported potentials decaying sufficiently fast and outlines how the critical set structure persists under suitable perturbations, setting the stage for broader applications and future work on the planar RC3BP case with slower decay. Overall, the paper provides a rigorous variational and homological toolkit to certify reachability via two boosts in noncompact celestial mechanics models and identifies precise topological invariants governing such connections.

Abstract

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems related to the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The main technical work goes into dealing with the noncompactness of the corresponding energy hypersurfaces.

Paper Structure

This paper contains 19 sections, 43 theorems, 274 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lagrangian Rabinowitz Floer homology
  3. The Lagrangian Rabinowitz action functional
  4. Grading
  5. Floer trajectories
  6. Defining Lagrangian Rabinowitz Floer homology
  7. Positive Lagrangian Rabinowitz Floer homology
  8. Bounds on the Floer trajectories
  9. The bounds on the action
  10. The set of infinitesimally small action derivation
  11. The $L^2$ bounds
  12. The maximum principle
  13. The $L^\infty$ bounds
  14. The Lagrangian Rabinowitz Floer homology of the Copernican Hamiltonian
  15. Positive Lagrangian Rabinowitz Floer homology
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $q_0,q_1\in \mathbb{R}^2$ and let $H_0$ be as in DefH0. For every $h \in \mathcal{H}$ the Lagrangian Rabinowitz Floer homology of $\mathcal{A}^{H_0-h}_{q_0,q_1}$ is well defined. Moreover, for any two $h_0,h_1\in \mathcal{H}$ the Lagrangian Rabinowitz Floer homology of $\mathcal{A}^{H_0-h_0}_{q_

Figures (4)

  • Figure 1: The three functions $f$ corresponding to energy $c=\frac{1}{5}$ (blue), $c=\frac{1}{10}$ (green), and $c=\frac{1}{20}$ (magenta) crossing zero in exactly $1$, $3$, and $5$ points, respectively.
  • Figure 2: The unique Reeb chords of energy $c=1$ (green), $c=\frac{1}{2}$ (blue), and $c=\frac{1}{5}$ (magenta).
  • Figure 3: The three Reeb chords of energy $c=\frac{1}{10}$.
  • Figure 4: The five Reeb chords of energy $c=\frac{1}{20}$.

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • ...and 79 more