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arxiv: 2607.00478 · v1 · pith:UAAULP4Cnew · submitted 2026-07-01 · 🧮 math.DS · math-ph· math.MP

Periodic orbits with prescribed negative energy for relativistic Keplerian problems

Alberto Boscaggin , Guglielmo Feltrin , Duccio Papini This is my paper

Pith reviewed 2026-07-02 05:33 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MP
keywords periodic orbitsrelativistic Kepler problemvariational methodsMaupertuis functionalnegative energyblow-up analysispenalization
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The pith

Periodic solutions exist on every prescribed negative energy level for the relativistic Kepler problem in dimensions two and higher.

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

The paper establishes the existence of periodic solutions with any prescribed negative energy for a relativistic Kepler equation perturbed by a lower-order potential term. The central difficulty is that the Kepler singularity makes the associated Maupertuis functional lack compactness exactly at the boundary between weak-force and strong-force regimes. The authors overcome this by a penalization procedure, a min-max scheme, and blow-up analysis of near-collision sequences, yielding non-perturbative existence results that hold in every dimension N at least 2.

Core claim

For the relativistic equation d/dt (m ˙x / sqrt(1 - |˙x|^2/c^2)) = -α x/|x|^3 + ∇W(x) with W a lower-order perturbation of the Kepler potential, periodic solutions exist on every prescribed negative energy level. This holds in all dimensions N ≥ 2 by a variational argument that restores compactness to the Maupertuis functional via penalization and blow-up analysis.

What carries the argument

Penalization procedure combined with min-max scheme and blow-up analysis applied to the Maupertuis functional.

If this is right

  • Periodic orbits exist for every negative energy level without requiring the perturbation to be small.
  • The variational method applies uniformly in every dimension N ≥ 2.
  • Solutions are obtained directly on the prescribed energy surface rather than by continuation from the unperturbed case.
  • The approach handles the critical singularity without restricting to weak or strong force regimes.

Where Pith is reading between the lines

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

  • The same compactness-recovery strategy could extend to other singular potentials whose strength places them at the weak-strong boundary.
  • Numerical integration of sample orbits at fixed negative energy would provide an independent check on the existence result.
  • The non-perturbative character suggests the result remains valid for perturbations whose size is comparable to the Kepler term at moderate distances.

Load-bearing premise

The penalization and blow-up analysis restore compactness to the Maupertuis functional even though the Kepler singularity sits exactly at the boundary between weak-force and strong-force regimes.

What would settle it

Explicit construction of a perturbation W and negative energy level E for which no periodic solution exists in dimension N=2, or numerical evidence that critical sequences fail to converge after penalization.

read the original abstract

Using a variational approach, we study the existence of periodic solutions with prescribed energy for the relativistic equation \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{m\dot x}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -\alpha \frac{x}{|x|^{3}} + \nabla W(x), \qquad x\in\mathbb{R}^{N}\setminus\{0\}, \end{equation*} where $W$ is a lower-order perturbation of the Kepler potential. The main difficulty stems from the fact that the Kepler singularity is critical for the associated Maupertuis functional, lying exactly at the boundary between the weak force and strong force regimes. To overcome the resulting lack of compactness, we use a penalization procedure and develop a suitable min-max scheme combined with a blow-up analysis of near-collision critical sequences. As a consequence, we establish the existence of periodic solutions on prescribed negative energy levels, obtaining non-perturbative results in every dimension $N\geq 2$.

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 / 1 minor

Summary. The paper proves existence of periodic solutions with prescribed negative energy for the relativistic Kepler problem with lower-order perturbation W in dimensions N≥2. It employs a variational approach on the Maupertuis functional, using a penalization procedure to restore compactness at the critical Kepler singularity (boundary between weak- and strong-force regimes), followed by a min-max construction and blow-up analysis of Palais-Smale sequences.

Significance. If the blow-up analysis succeeds in producing collision-free limits on the exact prescribed energy level, the result supplies non-perturbative existence theorems for a technically difficult critical-singularity case in relativistic dynamics, extending known results beyond perturbative regimes.

major comments (2)
  1. [blow-up analysis section] The central claim requires that the blow-up limit of penalized critical sequences satisfies the unpenalized relativistic equation, remains collision-free, and lies exactly on the prescribed negative energy level. The relativistic momentum term (with Lorentz factor) may change the scaling or energy identity relative to the Newtonian case; the manuscript must verify this identity explicitly after blow-up (see the paragraph on main difficulty and the blow-up analysis section).
  2. [penalization procedure] The penalization term is asserted to push solutions away from x=0 while preserving the min-max value corresponding to the target energy; it is not clear from the construction whether the penalization parameter can be sent to zero without shifting the energy level or introducing collisions in the limit (abstract and main difficulty paragraph).
minor comments (1)
  1. [introduction] Notation for the relativistic kinetic term and the precise form of the Maupertuis functional should be stated explicitly in the introduction for readers unfamiliar with the relativistic setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to provide the requested explicit verifications.

read point-by-point responses

  1. Referee: [blow-up analysis section] The central claim requires that the blow-up limit of penalized critical sequences satisfies the unpenalized relativistic equation, remains collision-free, and lies exactly on the prescribed negative energy level. The relativistic momentum term (with Lorentz factor) may change the scaling or energy identity relative to the Newtonian case; the manuscript must verify this identity explicitly after blow-up (see the paragraph on main difficulty and the blow-up analysis section).

    Authors: We agree that an explicit verification of the energy identity after blow-up is essential, especially given the relativistic momentum term. While our blow-up analysis is constructed to recover the unpenalized equation and preserve the energy level, we will add a dedicated computation in the revised blow-up analysis section that derives the limit energy identity step-by-step, confirming that the Lorentz factor does not alter the prescribed negative energy or introduce collisions. revision: yes

  2. Referee: [penalization procedure] The penalization term is asserted to push solutions away from x=0 while preserving the min-max value corresponding to the target energy; it is not clear from the construction whether the penalization parameter can be sent to zero without shifting the energy level or introducing collisions in the limit (abstract and main difficulty paragraph).

    Authors: We will expand the penalization procedure section (and the main difficulty paragraph) to clarify the choice of the penalization parameter and its limit. The revised text will include estimates showing that the min-max value is independent of the penalization parameter for sufficiently small values and that the limiting solutions remain collision-free on the exact target energy level. revision: yes

Circularity Check

0 steps flagged

Variational existence proof is self-contained; no circular reductions

full rationale

The paper establishes existence of periodic orbits via a penalization procedure, min-max scheme, and blow-up analysis applied to the Maupertuis functional for the relativistic Kepler problem. This is a standard functional-analytic construction that does not fit parameters to data, rename known results, or rely on self-citations whose content reduces to the target claim. The central result is an existence theorem whose proof steps (penalization to restore compactness, passage to the limit) are independent of the final statement and do not reduce by construction to the inputs. No load-bearing self-citation chains or self-definitional loops appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard variational calculus for singular potentials and on the specific penalization construction introduced to handle the critical singularity; no free parameters or new physical entities are introduced.

axioms (2)
  • ad hoc to paper The Maupertuis functional associated with the relativistic equation is amenable to a penalization procedure that restores compactness
    Invoked to overcome the lack of compactness caused by the critical Kepler singularity.
  • domain assumption Blow-up analysis of near-collision sequences yields a limit problem whose solutions can be ruled out or controlled
    Used after the min-max step to pass to the limit.

pith-pipeline@v0.9.1-grok · 5719 in / 1242 out tokens · 21155 ms · 2026-07-02T05:33:53.540043+00:00 · methodology

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Works this paper leans on

18 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Ambrosetti, V

    A. Ambrosetti, V. CotiZelati, Periodic solutionsof singularLagrangian systems,vol. 10 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, MA, 1993

    1993

  2. [2]

    C. M. Andersen, H. C. von Baeyer, On classical scalar field theories and the relativistic Kepler problem, Ann. Physics 62 (1971) 120–134

    1971

  3. [3]

    Arcoya, C

    D. Arcoya, C. Sportelli, Relativistic equations with singular potentials, Z. Angew. Math. Phys. 74 (2023) Paper No. 91, 22 pp

    2023

  4. [4]

    Bereanu, Interactions in the Lorentz force equation, Calc

    C. Bereanu, Interactions in the Lorentz force equation, Calc. Var. Partial Differential Equations 63 (2024) Paper No. 29, 25 pp

    2024

  5. [5]

    Boscaggin, W

    A. Boscaggin, W. Dambrosio, G. Feltrin, Periodic solutions to a perturbed relativistic Kepler problem, SIAM J. Math. Anal. 53 (2021) 5813–5834

    2021

  6. [6]

    Boscaggin, W

    A. Boscaggin, W. Dambrosio, G. Feltrin, Prescribed energy periodic solutions of Kepler problems with relativistic corrections, in: Topological methods for delay and ordinary differential equations with applications to continuum mechanics, vol. 51 of Adv. Mech. Math., Birkhäuser/Springer, Cham, 2024, pp. 21–41. Fixed energy relativistic Keplerian orbits 27

    2024

  7. [7]

    Boscaggin, W

    A. Boscaggin, W. Dambrosio, E. Muñoz Hernández, A Maupertuis-type principle in relativistic mechanics and applications, Calc. Var. Partial Differential Equations 62 (2023) Paper No. 95, 29 pp

    2023

  8. [8]

    Boscaggin, W

    A. Boscaggin, W. Dambrosio, D. Papini, Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential, J. Differential Equations 383 (2024) 190–213

    2024

  9. [9]

    Boscaggin, W

    A. Boscaggin, W. Dambrosio, D. Papini, Periodic solutions to relativistic Kepler prob- lems: a variational approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (2024) 1465–1504

    2024

  10. [10]

    Boscaggin, G

    A. Boscaggin, G. Feltrin, D. Papini, Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems, Calc. Var. Partial Differential Equations 64 (2025) Paper No. 64, 29 pp

    2025

  11. [11]

    Boscaggin, G

    A. Boscaggin, G. Feltrin, D. Papini, Bifurcation from periodic solutions of central force problems in the three-dimensional space, J. Differential Equations 470 (2026) Paper No. 114423, 34 pp

    2026

  12. [12]

    T. H. Boyer, Unfamiliar trajectories for a relativistic particle in a Kepler or Coulomb potential, Amer. J. Phys. 72 (2004) 992–997

    2004

  13. [13]

    A. C. Gallo, Periodic solutions of perturbed central Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl. 26 (2019) Paper No. 34, 24 pp

    2019

  14. [14]

    M. W. Hirsch, Differential topology, vol. No. 33 of Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976

    1976

  15. [15]

    Jia, Approximate Kepler’s elliptic orbits with the relativistic effects, Int

    L. Jia, Approximate Kepler’s elliptic orbits with the relativistic effects, Int. J. Astron. Astrophys. 3 (2013) 29–33

    2013

  16. [16]

    T. J. Lemmon, A. R. Mondragon, Kepler’s orbits and Special Relativity in Introductory Classical Mechanics, arXiv:1012.5438

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Muñoz, I

    G. Muñoz, I. Pavic, A Hamilton-like vector for the special-relativistic Coulomb prob- lem, European J. Phys. 27 (2006) 1007–1018

    2006

  18. [18]

    Giuseppe Peano

    K. Tanaka, A prescribed energy problem for a singular Hamiltonian system with a weak force, J. Funct. Anal. 113 (1993) 351–390. Alberto Boscaggin Dipartimento di Matematica “Giuseppe Peano” Università degli Studi di Torino Via Carlo Alberto 10, 10123 Torino, Italy e-mail:alberto.boscaggin@unito.it Guglielmo Feltrin Dipartimento di Scienze Matematiche, Inf...

    1993