The two-boost problem and the boundedness of Reeb chords

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.

💡 Research Summary

The paper addresses the “two‑boost problem” in space mission design, which asks whether two prescribed positions in configuration space can be linked by a Hamiltonian trajectory that lies on a fixed energy level and uses thrust only at the beginning and end of the journey. The authors focus on a class of Hamiltonian systems that, at infinity, behave like the restricted circular planar three‑body problem (RC3BP). Their strategy is to translate the existence question into a problem about Reeb chords for the Rabinowitz action functional and then to apply Lagrangian Rabinowitz Floer homology (LRFH).

First, the authors recall the classical Hohmann transfer for the Kepler problem, where any two circular orbits can be connected by two impulsive burns. They then formulate the two‑boost problem for a general Hamiltonian H on the cotangent bundle T*Q, with Q = ℝ² \ {collision points}. A chord (v, η) satisfies ∂ₜv = η X_H(v), stays on the energy hypersurface H⁻¹(c), and meets the cotangent fibers over the two prescribed points. Such chords are precisely the critical points of the Rabinowitz action functional
A_{H‑c}^{q₀,q₁}(v,η) = ∫₀¹ λ(∂ₜv) dt – η ∫₀¹ (H – c)(v(t)) dt.

The key technical tool is Lagrangian Rabinowitz Floer homology, which counts these critical points in a homological framework and is invariant under compact perturbations of the Hamiltonian. In a previous work