pith. sign in

arxiv: 2604.10559 · v4 · pith:FSAAK2KZnew · submitted 2026-04-12 · 🧮 math-ph · math.MP

Periods of N-body Systems Determined Through Dimensional Analysis

Dan Jonsson This is my paper

Pith reviewed 2026-05-21 01:09 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords dimensional analysisn-body problemKepler's third laworbital periodsgravitational systemsquantum mechanicsscaling laws
0
0 comments X

The pith

Generalized dimensional analysis determines the periods of n-body systems up to a multiplicative constant.

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

The paper shows that a generalization of classical dimensional analysis can be applied to gravitational n-body systems to find expressions for their orbital periods. This approach recovers the scaling behavior of Kepler's third law for two bodies and extends it to arbitrary numbers of bodies. The same technique produces a corresponding result for the quantum version of the problem. A reader would care because the method supplies the period scaling directly from the variables and constants involved, without needing to integrate the equations of motion.

Core claim

A generalization of classical dimensional analysis makes it possible to derive the period of a classical n-body system, up to a multiplicative constant, by the same technique used for Kepler's third law. The resulting expression agrees with the conjecture of Sun. The identical procedure applied to the quantum-theoretical counterpart of the classical n-body problem yields a period that agrees with the conjecture of Semay and Sun.

What carries the argument

The generalization of classical dimensional analysis that determines the period up to a multiplicative constant when applied to n-body gravitational systems.

If this is right

  • The period of a classical n-body system scales with total mass and characteristic size in a form that generalizes Kepler's third law.
  • The same scaling form holds for the quantum-theoretical counterpart of the classical n-body system.
  • Periods can be obtained without solving the differential equations that govern the motion.
  • The results confirm the functional dependence conjectured earlier by Sun for classical systems and by Semay and Sun for quantum systems.

Where Pith is reading between the lines

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

  • The method could be tested on observed periods in star clusters or planetary systems to see if the unknown multiplicative factor remains roughly constant across different n.
  • Similar dimensional-analysis shortcuts might apply to other long-range potentials or to systems with additional conserved quantities.
  • In the quantum setting the derived period could be compared with energy-level spacings in few-body atomic or molecular calculations.

Load-bearing premise

The generalization of classical dimensional analysis presented in a separate article applies directly to n-body systems.

What would settle it

Numerical integration of the equations of motion for a specific three- or four-body system with chosen masses and initial distances would show whether the observed period matches the scaling predicted by the dimensional analysis.

read the original abstract

A generalization of classical dimensional analysis, presented in a separate article, makes it possible to derive Kepler's third law for the period of a two-body system, up to a multiplicative constant, without solving the equations of motion. Here we show how to derive generalizations of Kepler's third law to n-body systems by the same technique. Our results agree with conjectures by Sun on the period of a classical n-body system and by Semay and Sun on the quantum-theoretical counterpart of the period of a classical n-body system.

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 manuscript claims that a generalization of classical dimensional analysis, introduced in a separate article, can be applied to derive generalizations of Kepler's third law for the periods of classical and quantum n-body gravitational systems. These derivations are obtained without solving the equations of motion and yield the periods up to an undetermined multiplicative constant; the resulting functional forms are stated to agree with conjectures by Sun (classical) and by Semay and Sun (quantum).

Significance. If the generalized technique applies without hidden assumptions about the governing equations, the work would offer a compact route to period scalings for multi-particle systems with 1/r potentials. This could be useful for identifying scaling behaviors in celestial mechanics and their quantum analogs, though the results remain dependent on an external multiplicative constant and on the validity of the prior generalization.

major comments (2)
  1. [Abstract and main derivation] The derivation invokes the generalized dimensional analysis directly from the separate article without re-deriving the dimensionless combinations or addressing how the multi-particle 1/r potential and 3n-6 internal degrees of freedom alter the dimensional basis. This is load-bearing for the central claim, as any restriction in the prior technique on the form of the equations would propagate unchanged into the n-body results.
  2. [Results and discussion] The agreement with Sun’s and Semay-Sun’s conjectures is presented as corroboration, yet the manuscript supplies no independent verification or error analysis once the multiplicative constant is fixed externally; this leaves the functional-form claim dependent on the external input rather than self-contained.
minor comments (1)
  1. Clarify in the text how the multiplicative constant is to be determined in practice for specific n-body configurations, as its undetermined status is noted but not explored.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below, indicating where we agree and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and main derivation] The derivation invokes the generalized dimensional analysis directly from the separate article without re-deriving the dimensionless combinations or addressing how the multi-particle 1/r potential and 3n-6 internal degrees of freedom alter the dimensional basis. This is load-bearing for the central claim, as any restriction in the prior technique on the form of the equations would propagate unchanged into the n-body results.

    Authors: The generalized dimensional analysis is presented in the companion paper as a general framework applicable to any system whose governing quantities have well-defined dimensions, including gravitational n-body problems with 1/r potentials. The 3n-6 internal degrees of freedom for n>2 are handled by choosing the characteristic length and time scales that describe the overall system size and dynamics, consistent with the standard treatment of the n-body problem. We did not repeat the full derivation of dimensionless combinations to keep the present manuscript focused and concise. However, we agree that a short paragraph summarizing the relevant dimensional basis for the multi-particle case would improve accessibility and address potential concerns about hidden assumptions. We will add this clarification in the revised version. revision: yes

  2. Referee: [Results and discussion] The agreement with Sun’s and Semay-Sun’s conjectures is presented as corroboration, yet the manuscript supplies no independent verification or error analysis once the multiplicative constant is fixed externally; this leaves the functional-form claim dependent on the external input rather than self-contained.

    Authors: The manuscript's central contribution is the derivation of the functional form of the period via generalized dimensional analysis, which by design determines the scaling up to a multiplicative constant (as is conventional in dimensional analysis, including the original Kepler's third law). The agreement with the conjectures of Sun (classical) and Semay-Sun (quantum) is offered as evidence that the technique correctly reproduces known results for these systems. The original conjectures were supported by numerical simulations and other analyses in the cited references, which provide the independent verification. Our work does not repeat those numerical checks, as that would duplicate prior efforts and fall outside the scope of demonstrating the dimensional method. We can add a brief sentence in the discussion section explicitly referencing the numerical support in the cited works to make this dependence clearer. revision: partial

Circularity Check

1 steps flagged

Central claim rests on unexamined applicability of generalized dimensional analysis from a separate article to n-body gravitational dynamics

specific steps
  1. self citation load bearing [Abstract]
    "A generalization of classical dimensional analysis, presented in a separate article, makes it possible to derive Kepler's third law for the period of a two-body system, up to a multiplicative constant, without solving the equations of motion. Here we show how to derive generalizations of Kepler's third law to n-body systems by the same technique."

    The n-body results are obtained by direct application of the generalized DA method from the separate article (likely by the same author) without independent re-derivation or addressing modifications for summed pair potentials and internal degrees of freedom; the claimed scaling therefore inherits its validity from the unexamined prior technique rather than standing on new analysis.

full rationale

The paper derives n-body period generalizations solely by invoking a generalized dimensional analysis technique introduced in a separate article, without re-deriving dimensionless combinations or validating the method for multi-particle 1/r potentials and 3n-6 degrees of freedom. This creates a load-bearing dependency on the prior work. While the functional forms are new and results match external conjectures, the core scaling (up to multiplicative constant) reduces to the applicability of the cited generalization rather than an independent derivation within this manuscript.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the generalized dimensional analysis from a separate paper to n-body systems and on the assumption that periods are determined solely by dimensional combinations of masses, distances, and G (or equivalent quantum constants) up to one overall factor.

free parameters (1)
  • multiplicative constant
    The period expressions are obtained only up to an overall multiplicative constant that is not fixed by the dimensional analysis itself.
axioms (1)
  • domain assumption The generalized dimensional analysis from the separate article applies without modification to n-body gravitational and quantum systems.
    Invoked in the abstract as the basis for extending the two-body result to arbitrary n.

pith-pipeline@v0.9.0 · 5596 in / 1339 out tokens · 70459 ms · 2026-05-21T01:09:22.820853+00:00 · methodology

Review history (3 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

  1. [1]

    Carroll, D.A

    B.W. Carroll, D.A. Ostlie. An introduction to modern astrophysics, 2nd ed. (2007)

    work page 2007

  2. [2]

    I. Newton. Philosophiæ Naturalis Principia Mathematica (1687)

    work page

  3. [3]

    Bernoulli, Extrait de la Reponse de M

    J. Bernoulli, Extrait de la Reponse de M. Bernoulli à M. Herman, datée de Basle le 7. Octobre

    work page

  4. [4]

    Royale des Sciences, Boudot Paris 1710 (1712)

    Memoires de l’Ac. Royale des Sciences, Boudot Paris 1710 (1712)

    work page

  5. [5]

    Euler L, Nov. Comm. Acad. Imp. Petropolitanae10, pp. 207–242,11, pp. 152–184; Mémoires de l’Acad. de Berlin11, 228–249

    work page

  6. [6]

    Lagrange

    J.L. Lagrange. Essai sur le problème des trois corps, Oeuvres, tome 6 (1772)

    work page

  7. [7]

    Poincaré

    H. Poincaré. Les Méthodes nouvelles de la mécanique céleste, Gauthier-Villars (1892, 1893, 1899)

    work page

  8. [8]

    Sun, Kepler’s third law ofn-body periodic orbits in a Newtonian gravitation field

    BH. Sun, Kepler’s third law ofn-body periodic orbits in a Newtonian gravitation field. Science China Physics, Mechanics and Astronomy61, 064721 (2018)

    work page 2018

  9. [9]

    X.M. Li, S.J. Liao. More than six hundred new families of Newtonian periodic, planar colli- sionless 3-body orbits. Science China Physics, Mechanics and Astronomy60, 129512 (2017)

    work page 2017

  10. [10]

    The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum

    Xiaoming Li, Yipeng Jing, Shijun Liao. The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum. arXiv:1709.04775 (2017)

    work page arXiv 2017

  11. [11]

    D. Jonsson. Quantities, Dimensions, and Dimensional Analysis, arXiv:1408.5024 (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  12. [12]

    D. Jonsson. On Augmented Dimensional Analysis. Acta Appl Math 192:10; https://doi.org/10.1007/s10440-024-00673-y (2024)

    work page doi:10.1007/s10440-024-00673-y 2024

  13. [13]

    C.Semay.QuantumsupporttoBoHuaSun’sconjecture.ResultsinPhysics13, 102167(2019)

    work page 2019

  14. [14]

    Semay, C.T

    C. Semay, C.T. Willemyns. Quasi Kepler’s third law for quantum many-body systems. Eur. Phys. J. Plus 136:342 (2021)

    work page 2021

  15. [15]

    BH. Sun. Classical and quantum Kepler’s third law on N-body system. Results in Physics 13, 102144 (2019)

    work page 2019

  16. [16]

    Montgomery, The N-body problem, the braid group, and action-minimizing periodic so- lutions, Nonlinearity11, 363–376 (1998)

    R. Montgomery, The N-body problem, the braid group, and action-minimizing periodic so- lutions, Nonlinearity11, 363–376 (1998)

    work page 1998

  17. [17]

    Topological dependence of Kepler’s third law for collision- less periodic three-body orbits with vanishing angular momentum and equal masses

    Dmitrašinović, V., & Šuvakov, M. Topological dependence of Kepler’s third law for collision- less periodic three-body orbits with vanishing angular momentum and equal masses. Physics Letters A, 379(36), 1939–1945 (2015)

    work page 1939