The two-boost problem and the boundedness of Reeb chords

Abstract: The two-boost problem in space mission design asks whether two points of position space can be connected by a Hamiltonian path on a fixed energy level set. We provide a positive answer for a class of systems having the same behaviour at infinity as the restricted three-body problem by relating it to the Lagrangian Rabinowitz Floer homology computed in [4]. The main technical challenge is to prove the boundedness of the corresponding sets of Reeb chords.

• The paper establishes that two-boost Hamiltonian transfers exist and remain bounded under explicit energy thresholds in the RTBP framework.
• It employs a Hamiltonian and Rabinowitz Floer homological approach to control moduli space compactness and prevent trajectories from escaping.
• The results offer practical criteria for space mission design, demonstrating how symplectic methods enhance maneuver planning in celestial mechanics.

Summary of "The two-boost problem and the boundedness of Reeb chords" (2512.13373)

Problem Formulation and Motivation

The two-boost problem, relevant in space mission design, asks whether it is possible to connect two points in position space by a Hamiltonian path within a fixed energy level, using impulsive engine boosts only at the trajectory’s start and end. The classic solution—a Hohmann transfer in the Kepler problem—extends this question to scenarios governed by more complex dynamics, particularly the planar restricted three-body problem (RTBP), where the non-compactness of the phase space arises due to possible collisions and escape to infinity.

Wiśniewska’s paper analyzes whether two points within a specified compact subset of $\mathbb{R}^2$ can be connected by such a trajectory in the RTBP setting, focusing on the behavior at infinity rather than collision-induced singularities, which are addressable via Birkhoff regularization.

Hamiltonian Framework and Floer Homological Approach

The dynamics are studied on $T^*Q$, the cotangent bundle of the position manifold $Q$, with a Hamiltonian $H: T^*Q \to \mathbb{R}$. The central mathematical object is the existence of Hamiltonian "chords"—paths in phase space at fixed energy connecting two specified cotangent fibers. These chords are formally characterized as critical points of the Lagrangian Rabinowitz action functional

$$A^{H-c}_{q_0,q_1}(v, \eta) = \int_0^1 \lambda(\partial_t v)\,dt - \eta \int_0^1 (H-c)(v(t))\,dt$$

with boundary conditions $v(0), v(1) \in T^*_{q_0}Q, T^*_{q_1}Q$.

Building on prior work that generalized Rabinowitz Floer homology to non-compact settings, the paper studies Hamiltonians that outside a compact set coincide with the RTBP Hamiltonian,

$H(q, p) = H_0(q, p) - V(q)$

where $H_0$ is the kinetic plus Coriolis part, and $V$ captures the gravitational pull of the primaries. The primary technical challenge is controlling the compactness properties of the moduli space of Reeb chords—specifically, ensuring solutions are bounded and cannot escape to infinity when energy is sufficiently high.

Main Results

The paper establishes that for Hamiltonians of the above type, two points in $\mathbb{R}^2$ lying inside a prescribed compact ball $B(R_1)$ can always be connected by a Hamiltonian path—i.e., a two-boost transfer—if either:

• The energy level $c$ is chosen above a specific threshold (explicit formulas given), or
• The potential $V$ is rotationally invariant outside $B(R_1)$ and $c$ exceeds a different threshold.

Crucially, all solutions are guaranteed to remain within the region $T^*B(R_1)$, precluding nonphysical escape at high energies. The results are formalized in Theorems detailing precise energy bounds and regularity assumptions.

A notable application is to the regularized RTBP: for the class of Hamiltonians that agree with the RTBP Hamiltonian outside a compact subset, the paper provides explicit criteria guaranteeing the existence of connecting chords between any two points in $B(R_1)$ at sufficiently large values of $c$. The analysis is intricate, involving estimates mimicking the asymptotic hyperbolic behavior of the (rotating) Kepler problem at infinity.

Technical Innovations

Key technical contributions include:

• Extension and computation of Lagrangian Rabinowitz Floer homology for Hamiltonians with compact perturbations in non-compact settings (referencing earlier work by Wiśniewska et al., particularly [Wisniewska2024]).
• A reduction of the two-boost existence question to the non-vanishing of said homology and the boundedness of associated moduli spaces of chords.
• Detailed estimates (lemmas and propositions) proving that for high energies, Hamiltonian trajectories outside compact sets exhibit hyperbolic behavior, guaranteeing trajectories do not return from infinity.
• Construction of perturbed Hamiltonians $H_1$ that allow control over support and regularity, facilitating compactness arguments.

Strong Numerical and Conceptual Claims

Quantitative criteria for energy bounds (expressed in explicit inequalities involving system parameters) are provided, ensuring safe regimes for two-point connections. Additionally, for potentials satisfying rotational invariance at infinity, the preservation of angular momentum strengthens boundedness guarantees.

Implications and Prospects

These results have direct implications for space mission planning under RTBP-like dynamics, confirming the theoretical attainability of two-point transfers with only initial and final impulses under realistic energy constraints. From a mathematical perspective, the work demonstrates how symplectic and Floer-theoretic techniques can be adapted to global dynamical problems with non-compact phase spaces.

Future work may focus on incorporating collision singularities post-regularization, refining energy bounds, and extending these results to fully general three-body or higher-dimensional celestial mechanics, exploiting advances in symplectic topology and Floer theory. Dynamical systems exhibiting similar asymptotics should also be amenable to this framework.

Conclusion

Wiśniewska’s analysis provides rigorous, computable criteria guaranteeing the existence and boundedness of two-boost trajectories in generalized RTBP contexts, leveraging advanced tools from Rabinowitz Floer homology and symplectic geometry. This formalizes and extends a classic question in mission design, opening the door to further work on global connections in non-compact Hamiltonian systems and their applications in celestial mechanics.