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arxiv: 2512.13373 · v2 · pith:5XFAOLW6new · submitted 2025-12-15 · 🧮 math.SG

Two-boost problem for the Newtonian potential at the infinity

Jagna Wi\'sniewska This is my paper

Pith reviewed 2026-05-25 07:14 UTC · model grok-4.3

classification 🧮 math.SG
keywords two-boost problemNewtonian potentialrestricted three-body problemLagrangian Rabinowitz Floer homologyReeb chordsHamiltonian pathsenergy level setssymplectic geometry
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The pith

The two-boost problem admits positive solutions for systems behaving like the restricted three-body problem at infinity.

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

The two-boost problem asks whether any two points in position space can be joined by a Hamiltonian trajectory that remains on a fixed energy level. The paper gives an affirmative answer for systems whose large-distance dynamics match those of the restricted three-body problem. The argument proceeds by proving that the sets of Reeb chords are bounded, which lets the problem reduce to a previously computed Lagrangian Rabinowitz Floer homology group. A positive result means that connecting paths exist on the given energy surface for this class of Newtonian systems.

Core claim

For systems with the same behavior at infinity as the restricted three-body problem, the two-boost problem has a positive answer because the boundedness of Reeb chord sets permits direct application of the Lagrangian Rabinowitz Floer homology computation from prior work, establishing the existence of the required Hamiltonian paths.

What carries the argument

Lagrangian Rabinowitz Floer homology, applied after a proof that the sets of Reeb chords remain bounded on the energy surface.

If this is right

  • Connecting Hamiltonian paths exist on the fixed energy level for every pair of points in position space.
  • The boundedness result extends the reach of the prior homology computation to a wider family of systems.
  • Non-vanishing of the homology group implies solutions to the two-boost problem whenever the chord sets stay bounded.
  • The same reduction applies to any Hamiltonian system whose asymptotic dynamics coincide with those of the restricted three-body problem.

Where Pith is reading between the lines

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

  • The boundedness technique may transfer to other celestial-mechanics problems that preserve the same infinity behavior.
  • Explicit trajectories could now be located by combining the homology existence result with numerical shooting methods.
  • The argument indicates that symplectic invariants remain effective for detecting connections when only the far-field potential is controlled.

Load-bearing premise

The systems under study must share the same behavior at infinity as the restricted three-body problem so the earlier homology computation applies directly.

What would settle it

An explicit system with matching infinity behavior for which the sets of Reeb chords on the energy surface are unbounded, or a direct calculation showing the relevant homology vanishes.

read the original 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.

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

1 major / 0 minor

Summary. The manuscript claims a positive answer to the two-boost problem (existence of a Hamiltonian path on a fixed energy level connecting two points in position space) for a class of systems sharing the same asymptotic behavior at infinity as the restricted three-body problem. The argument proceeds by establishing boundedness of the relevant sets of Reeb chords and then reducing the existence question to the Lagrangian Rabinowitz Floer homology computation already obtained in reference [4].

Significance. If the boundedness step is carried out rigorously and the reduction to [4] is valid, the result would extend existence theorems for connecting orbits from the restricted three-body problem to a larger family of Hamiltonian systems arising in celestial mechanics and space-mission design. The high-level strategy is logically coherent and leverages an existing homology computation, which is a strength when the adaptation is justified.

major comments (1)
  1. [Abstract] Abstract: the central claim is a positive answer obtained by relating the two-boost problem to the homology computation in [4] after proving boundedness of Reeb chords; however, the abstract supplies no proof outline, error estimates, or verification steps for the boundedness argument, so it is impossible to assess whether the reduction actually supports the claim without the full manuscript details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for greater clarity in the abstract. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is a positive answer obtained by relating the two-boost problem to the homology computation in [4] after proving boundedness of Reeb chords; however, the abstract supplies no proof outline, error estimates, or verification steps for the boundedness argument, so it is impossible to assess whether the reduction actually supports the claim without the full manuscript details.

    Authors: We agree that the abstract, as currently written, is too concise and does not outline the boundedness argument. The full manuscript contains the detailed proof that the relevant sets of Reeb chords are bounded (via the asymptotic behavior matching the restricted three-body problem), which justifies the reduction to the Lagrangian Rabinowitz Floer homology computation of [4]. In the revised version we will expand the abstract by one or two sentences that sketch the key steps of the boundedness proof, while keeping the abstract within journal length limits. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central argument proceeds by proving boundedness of the sets of Reeb chords (identified as the main technical step) for systems sharing the asymptotic behavior at infinity with the restricted three-body problem, after which the Lagrangian Rabinowitz Floer homology result from [4] applies directly. This structure uses an external prior computation as a black-box tool once the new boundedness condition is verified; the derivation does not reduce any claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The provided abstract and high-level strategy contain no equations or steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on the domain assumption that the systems match the restricted three-body problem at infinity and on the prior homology computation; no free parameters or invented entities appear in the abstract.

axioms (1)
  • domain assumption Systems have the same behaviour at infinity as the restricted three-body problem
    Explicitly stated as the class of systems for which the positive answer holds.

pith-pipeline@v0.9.0 · 5585 in / 1093 out tokens · 40865 ms · 2026-05-25T07:14:07.411344+00:00 · methodology

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Reference graph

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