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arxiv: 2607.00517 · v1 · pith:MJCVX6WZnew · submitted 2026-07-01 · 🧮 math.DS

A Minimax Approach to Relative Periodic Orbits in Symmetric Three-Degree-of-Freedom Hamiltonian Systems

Shu Sakaguchi , Mitsuru Shibayama This is my paper

Pith reviewed 2026-07-02 05:31 UTC · model grok-4.3

classification 🧮 math.DS
keywords relative periodic orbitsHamiltonian systemsminimax methodMaupertuis functionalHill regionsthree-body problemKepler problem
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The pith

Under topological assumptions on compact Hill regions, a minimax method on the Maupertuis functional yields periodic solutions on every energy surface of the reduced two-degree-of-freedom system.

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

The paper examines three-degree-of-freedom Hamiltonian systems that remain unchanged under rotation about the z-axis and reflection across the xy-plane. Angular momentum is fixed to reduce the system to two degrees of freedom. On ranges of energy where the Hill regions stay compact, suitable topological assumptions allow a variational minimax argument to locate periodic solutions as saddle points of the Maupertuis functional. These solutions are either nontrivial spatial periodic orbits or trivial planar brake orbits in the reduced system. A Morse-index condition is given that guarantees the solutions are nontrivial, and the results are checked on the isosceles three-body problem and the spatial anisotropic Kepler problem.

Core claim

Under suitable assumptions on the topology of these compact Hill regions, periodic solutions exist on each prescribed energy surface of the reduced system as saddle points of the Maupertuis functional. The resulting solutions are either nontrivial spatial periodic solutions or trivial planar brake solutions in the reduced system. Computing the Morse index supplies a sufficient condition that ensures the periodic solutions obtained are nontrivial.

What carries the argument

The Maupertuis functional defined on the energy surfaces of the reduced two-degree-of-freedom system, with a minimax procedure locating its saddle points.

Load-bearing premise

The topology of the compact Hill regions permits the minimax method to be applied to the Maupertuis functional on each energy surface.

What would settle it

An explicit energy value and Hill region whose topology satisfies the stated assumptions yet whose Maupertuis functional possesses only the trivial planar critical points.

read the original abstract

We study three-degree-of-freedom Hamiltonian systems that are invariant under rotations about the $z$-axis and under reflection across the $xy$-plane. Fixing the angular momentum, such systems reduce to Hamiltonian systems with two degrees of freedom. We focus on the range of energy values for which the corresponding Hill regions are compact. First, under suitable assumptions on the topology of these compact Hill regions, we prove the existence of periodic solutions on each prescribed energy surface of the reduced system by means of a variational minimax method. These periodic solutions are obtained as saddle points of the Maupertuis functional. The resulting solutions are either nontrivial spatial periodic solutions or trivial planar brake solutions in the reduced system. Next, by computing the Morse index, we provide a sufficient condition ensuring that the periodic solutions obtained are nontrivial. Finally, we apply our results to the isosceles three-body problem and to the spatial anisotropic Kepler problem. In both cases, we verify the sufficient condition for nontriviality and thereby establish the existence of nontrivial periodic solutions.

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

0 major / 2 minor

Summary. The manuscript claims that, under suitable topological assumptions on compact Hill regions, a minimax argument applied to the Maupertuis functional yields periodic solutions on every energy surface of the reduced 2DOF Hamiltonian system obtained by fixing angular momentum in a 3DOF system invariant under z-axis rotations and xy-plane reflections. These critical points are either nontrivial spatial periodic orbits or trivial planar brake orbits; a Morse-index criterion is derived to guarantee nontriviality, and the criterion is verified explicitly for the isosceles three-body problem and the spatial anisotropic Kepler problem.

Significance. If the topological hypotheses hold and the index computations are correct, the work supplies a conditional variational existence theorem for relative periodic orbits in symmetric Hamiltonian systems, together with concrete nontriviality verifications in two standard celestial-mechanics examples. The approach combines established minimax and Morse-theory techniques in a manner that is directly applicable once the topological assumptions are checked.

minor comments (2)
  1. The topological assumptions required for the minimax theorem are referred to only as 'suitable' in the abstract and introduction; a concise, self-contained list of these hypotheses (e.g., connectedness, genus, or homology conditions on the Hill region) would improve readability.
  2. Notation for the reduced Maupertuis functional and the associated Sobolev space should be introduced once and used consistently; occasional shifts between the original 3DOF and reduced 2DOF variables can be confusing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were raised in the report, so we have no point-by-point rebuttals to provide. We will incorporate any minor editorial or typographical suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; standard conditional variational existence theorem

full rationale

The derivation proceeds by reducing the 3DOF symmetric Hamiltonian to a 2DOF system via angular momentum fixing, then applying a minimax argument to the Maupertuis functional on compact Hill regions whose topology satisfies stated hypotheses, yielding critical points that are either nontrivial spatial orbits or planar brake orbits. A separate Morse-index computation supplies a sufficient condition for nontriviality. Both the existence theorem and the index criterion are standard tools from variational calculus and Morse theory; the paper states the topological hypotheses explicitly, verifies them in the two applications (isosceles three-body and spatial anisotropic Kepler), and does not rely on fitted parameters, self-definitional relations, or load-bearing self-citations. The logical chain is self-contained against external mathematical benchmarks and contains no step that reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the central claim rests on topological assumptions about compact Hill regions and standard results from variational calculus and Morse theory. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Suitable assumptions on the topology of compact Hill regions allow application of the minimax method to the reduced system.
    Invoked in the first paragraph of the abstract to guarantee existence on each energy surface.
  • standard math The Maupertuis functional admits saddle points corresponding to periodic solutions under the stated symmetries.
    Core of the variational minimax method described.

pith-pipeline@v0.9.1-grok · 5712 in / 1265 out tokens · 23790 ms · 2026-07-02T05:31:46.973889+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references

  1. [1]

    Ambrosetti and V

    A. Ambrosetti and V. Coti Zelati.Periodic solutions of singular Lagrangian systems, volume 10 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1993

    1993

  2. [2]

    V. Benci. Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems.Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 1(5):401–412, 1984

    1984

  3. [3]

    Benci and F

    V. Benci and F. Giannoni. A new proof of the existence of a brake orbit. InAdvanced topics in the theory of dynamical systems (Trento, 1987), volume 6 ofNotes Rep. Math. Sci. Engrg., pages 37–49. Academic Press, Boston, MA, 1989

    1987

  4. [4]

    S. V. Bolotin. Libration motions of natural dynamical systems.Vestnik Moskov. Univ. Ser. I Mat. Mekh., (6):72–77, 1978

    1978

  5. [5]

    Chenciner and R

    A. Chenciner and R. Montgomery. A remarkable periodic solution of the three-body problem in the case of equal masses.Ann. of Math. (2), 152(3):881–901, 2000

    2000

  6. [6]

    Ghoussoub.Duality and perturbation methods in critical point theory

    N. Ghoussoub.Duality and perturbation methods in critical point theory. Number 107. Cambridge University Press, 1993

    1993

  7. [7]

    Gluck and W

    H. Gluck and W. Ziller. Existence of periodic motions of conservative systems. InSeminar on minimal submanifolds, volume 103 ofAnn. of Math. Stud., pages 65–98. Princeton Univ. Press, Princeton, NJ, 1983

    1983

  8. [8]

    J. L. G. Guirao, J. Llibre, and J. A. Vera. Periodic orbits of Hamiltonian systems: applications to perturbed Kepler problems.Chaos Solitons Fractals, 57:105–111, 2013

    2013

  9. [9]

    M. C. Gutzwiller. The anisotropic Kepler problem in two dimensions.J. Mathematical Phys., 14:139–152, 1973

    1973

  10. [10]

    K. Hayashi. Periodic solution of classical Hamiltonian systems.Tokyo J. Math., 6(2):473–486, 1983

    1983

  11. [11]

    X. Hu, Y. Ou, and Z. Qiao. Relative periodic orbits in the spatial anisotropic Kepler problem. Discrete Contin. Dyn. Syst., 52:434–447, 2026

    2026

  12. [12]

    Li and Q

    M. Li and Q. Liu. Periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations.Electron. J. Differential Equations, pages Paper No. 63, 42, 2021

    2021

  13. [13]

    Offin and H

    D. Offin and H. Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem.Discrete and Continuous Dynamical Systems-S, 2(2):379–392, 2009

    2009

  14. [14]

    P. H. Rabinowitz.Minimax methods in critical point theory with applications to differential equa- tions, volume 65 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. 20

    1986

  15. [15]

    Sakaguchi

    S. Sakaguchi. Periodic orbits in the spatial anisotropic Kepler problem: Existence and stability. Submitted

  16. [16]

    Shibayama

    M. Shibayama. Existence and stability of periodic solutions in the isosceles three-body problem. RIMS Kˆ okyˆ uroku Bessatsu B, 13:141–155, 2009. 21

    2009