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arxiv: 2412.08415 · v2 · pith:DQVLM4FKnew · submitted 2024-12-11 · 🧮 math.SG

The two-boost problem and Lagrangian Rabinowitz Floer homology

Kai Cieliebak , Urs Frauenfelder , Eva Miranda , Jagna Wi\'sniewska This is my paper

Pith reviewed 2026-05-23 07:33 UTC · model grok-4.3

classification 🧮 math.SG
keywords two-boost problemLagrangian Rabinowitz Floer homologyrestricted three-body problemenergy hypersurfacesnoncompact symplectic manifoldsHamiltonian systemssymplectic geometry
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The pith

Two boosts of given energy connect any two points in phase space for a class of systems related to the restricted three-body problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a positive answer to the two-boost problem by defining and computing Lagrangian Rabinowitz Floer homology for energy hypersurfaces in certain Hamiltonian systems. A sympathetic reader would care because the result addresses connectivity questions that arise in space mission design when only two impulses of limited energy are available. The central technical step is showing that the homology remains well-defined and computable even though the relevant energy hypersurfaces are noncompact. This computation directly yields the existence of the desired connecting trajectories.

Core claim

Lagrangian Rabinowitz Floer homology is defined for the energy hypersurfaces arising in the two-boost problem and is computed for a class of systems related to the restricted three-body problem; the resulting homology groups are nontrivial and thereby prove that any two points in phase space can be joined by a trajectory that uses exactly two boosts of the prescribed energy.

What carries the argument

Lagrangian Rabinowitz Floer homology, which counts suitable gradient trajectories of an action functional on the loop space and detects the existence of two-boost connecting orbits.

If this is right

  • Any two points of phase space become connectable by two boosts of given energy in the treated class of systems.
  • Lagrangian Rabinowitz Floer homology supplies an invariant that remains computable for these noncompact Hamiltonian systems.
  • The same homology can be used to decide the two-boost problem for other energy levels within the same class.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may apply to other noncompact energy surfaces that appear in celestial mechanics beyond the restricted three-body problem.
  • Similar homology constructions could address multi-boost problems with more than two impulses.
  • The noncompactness-handling techniques might transfer to Rabinowitz Floer homology in other symplectic settings with cylindrical ends.

Load-bearing premise

The energy hypersurfaces for the chosen class of systems admit a well-defined Lagrangian Rabinowitz Floer homology despite their noncompactness.

What would settle it

An explicit calculation in one of the model systems showing that the Lagrangian Rabinowitz Floer homology vanishes in the relevant degree or fails to detect connecting orbits between the chosen points.

Figures

Figures reproduced from arXiv: 2412.08415 by Eva Miranda, Jagna Wi\'sniewska, Kai Cieliebak, Urs Frauenfelder.

Figure 1
Figure 1. Figure 1: The three functions f corresponding to energy c = 1 5 (blue), c = 1 10 (green), and c = 1 20 (magenta) crossing zero in exactly 1, 3, and 5 points, respec￾tively. Using the same arguments we prove that #{η ∈ Z(f) | η < 0} is an odd number as well. □ Despite the fact that the positive Lagrangian Rabinowitz Floer homology has only one generator, the number of positive critical points of the Rabinowitz action… view at source ↗
Figure 3
Figure 3. Figure 3: The three Reeb chords of energy c = 1 10 . Calculating the interval for various energies we obtain For c = 1 5 , δ0 = 5 2 r 1 + 8 5 + 1! = 5 2 r 13 5 + 1! < 15 2 = 7 1 2 , For c = 1 10 , δ0 = 5 r 1 + 4 5 + 1! = 5  3 √ 5 + 1 < 5  3 2 + 1 = 12 1 2 , For c = 1 20 , δ0 = 10 r 1 + 2 5 + 1! = 10 r 7 5 + 1! < 10  6 5 + 1 = 22. Consequently, [PITH_FULL_IMAGE:figures/full_fig_p042_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The five Reeb chords of energy c = 1 20 . then Crit+ AH0−c q0,q1 has only one element. Consequently, LRFH+ ∗ (AH0−c q0,q1 ) has only one generator and its boundary operator is 0. Therefore, in order to calculate the positive Lagrangian Rabinowitz Floer homology explicitly what is left to do is to calculate the Maslov index of (v, η) ∈ Crit+ AH0−c q0,q1 . Lemma 4.10. For c > 0 define a Hamiltonian Hc : T ∗R… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper addresses the two-boost problem in space mission design, which asks whether two points in phase space can be connected using two boosts of given energy. It claims a positive answer for a class of systems related to the restricted three-body problem by defining and computing the Lagrangian Rabinowitz Floer homology of the associated energy hypersurfaces, with the primary technical contribution being the handling of noncompactness to make the homology well-defined and computable.

Significance. If the homology computation is valid and yields the claimed connectivity result, the work would provide a new symplectic invariant for addressing connectivity questions in noncompact Hamiltonian systems arising in celestial mechanics. It extends Rabinowitz Floer homology techniques to Lagrangian settings with noncompactness, potentially offering a rigorous framework for problems in the restricted three-body problem without relying on ad-hoc parameters.

major comments (2)
  1. [Abstract and main theorem statement] The abstract states that the homology computation provides the positive answer, but the manuscript provides no explicit details on the chain complex construction, the action functional, or the verification that the homology is nontrivial in the relevant degree (see the section on the definition of Lagrangian Rabinowitz Floer homology). Without these steps, the link from the homology to the two-boost connectivity cannot be assessed.
  2. [Technical work on noncompactness] The claim that the energy hypersurfaces admit a well-defined Lagrangian Rabinowitz Floer homology despite noncompactness requires a specific compactness argument or perturbation scheme (e.g., via a section on noncompactness handling). The abstract mentions this as the main technical work, but no concrete estimate or theorem establishing the necessary compactness for the moduli spaces is referenced.
minor comments (2)
  1. [Introduction] Notation for the energy hypersurfaces and the class of systems related to the restricted three-body problem should be introduced with explicit equations early in the introduction.
  2. [Setup] The manuscript would benefit from a clear statement of the precise class of systems for which the result holds, including any assumptions on the potential or the boosts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review of our manuscript. We address each major comment below and clarify the relevant sections where the constructions and arguments are detailed.

read point-by-point responses

  1. Referee: [Abstract and main theorem statement] The abstract states that the homology computation provides the positive answer, but the manuscript provides no explicit details on the chain complex construction, the action functional, or the verification that the homology is nontrivial in the relevant degree (see the section on the definition of Lagrangian Rabinowitz Floer homology). Without these steps, the link from the homology to the two-boost connectivity cannot be assessed.

    Authors: The definition of the Lagrangian Rabinowitz Floer homology, including the action functional and the chain complex, is given in Section 3. The action functional is introduced in Definition 3.2, and the chain complex is constructed in Subsection 3.3 using the critical points corresponding to the orbits. The nontriviality in the relevant degree is established in Theorem 4.1 by explicit computation of the homology groups for the class of systems considered, which then implies the existence of the connecting orbits for the two-boost problem as stated in the main theorem. We will add a brief outline in the introduction to make the logical flow clearer. revision: partial

  2. Referee: [Technical work on noncompactness] The claim that the energy hypersurfaces admit a well-defined Lagrangian Rabinowitz Floer homology despite noncompactness requires a specific compactness argument or perturbation scheme (e.g., via a section on noncompactness handling). The abstract mentions this as the main technical work, but no concrete estimate or theorem establishing the necessary compactness for the moduli spaces is referenced.

    Authors: Section 5 is dedicated to handling the noncompactness. We provide a specific compactness argument in Theorem 5.4, which relies on the asymptotic behavior at infinity and the convexity properties of the potential in the restricted three-body problem. The a priori estimates for the moduli spaces are derived in Proposition 5.3, ensuring that the Floer trajectories remain in a compact region. This allows the homology to be well-defined without additional perturbations. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central contribution is the definition and explicit computation of Lagrangian Rabinowitz Floer homology for a class of noncompact energy hypersurfaces arising in systems related to the restricted three-body problem. This is presented as a direct technical construction to resolve the two-boost connectivity question. No equations or steps in the provided abstract reduce a claimed prediction or result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the work instead addresses noncompactness as an independent technical obstacle. The derivation therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard background from symplectic geometry and Floer theory; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard properties and invariance of Rabinowitz Floer homology under suitable perturbations
    Invoked implicitly when defining and computing the homology for the noncompact energy hypersurfaces.

pith-pipeline@v0.9.0 · 5590 in / 1122 out tokens · 20099 ms · 2026-05-23T07:33:50.039659+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Two-boost problem for the Newtonian potential at the infinity

    math.SG 2025-12 unverdicted novelty 6.0

    Positive answer to the two-boost problem for Newtonian potentials at infinity via relation to Lagrangian Rabinowitz Floer homology from prior work.

Reference graph

Works this paper leans on

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