pith. sign in

arxiv: 2506.00086 · v1 · submitted 2025-05-30 · ⚛️ physics.class-ph

Does Newtonian dynamics need Euclidean space?

Alain Albouy This is my paper

Pith reviewed 2026-05-19 13:17 UTC · model grok-4.3

classification ⚛️ physics.class-ph
keywords Newtonian forceKepler's lawshomogeneous functionsgeneralized orbitshodographsconvexitycentral force problems
0
0 comments X

The pith

Newtonian force can be deduced from Kepler's laws by replacing Euclidean distance with any homogeneous function of the same degree.

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

The paper gives a direct derivation of the inverse-square force law starting from Kepler's three laws of planetary motion. It ties this derivation to an older generalization, due to Jacobi, that keeps the same scaling behavior but drops the specific Euclidean distance in the plane. The orbits in this wider setting still show the convexity properties familiar from the classical case, and their hodographs can be written down explicitly. If the argument holds, the force law itself does not require the full structure of Euclidean geometry.

Core claim

An elementary chain of steps extracts the Newtonian central force directly from the area law and the closed elliptical orbit. The same steps remain valid when the Euclidean form is exchanged for any other function homogeneous of degree one; the resulting orbits keep their convexity and the hodograph takes a simple geometric shape in the generalized setting.

What carries the argument

Replacement of the Euclidean distance by an arbitrary homogeneous function of the same degree, which leaves the key dynamical relations and convexity intact.

If this is right

  • The inverse-square law emerges from the area law and elliptical shape without invoking the full Euclidean metric.
  • Convexity of the generalized orbits follows from the same homogeneity condition.
  • Hodographs remain simple curves that can be described geometrically for every such homogeneous function.
  • Classical Keplerian properties survive the replacement of Euclidean distance by any equally homogeneous alternative.

Where Pith is reading between the lines

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

  • The same homogeneity argument might extend to other central-force problems whose potentials scale in a comparable way.
  • One could search for observable orbital signatures that would distinguish Euclidean space from other homogeneous alternatives at planetary scales.
  • Higher-dimensional versions or different degrees of homogeneity could uncover additional integrals of motion not visible in the plane.

Load-bearing premise

Any function homogeneous of the same degree as Euclidean distance will preserve the dynamical relations and convexity needed to recover the force law.

What would settle it

Explicit computation of the force for one concrete non-Euclidean homogeneous function that produces non-closed orbits or a hodograph that deviates from the predicted generalized form.

Figures

Figures reproduced from arXiv: 2506.00086 by Alain Albouy.

Figure 1
Figure 1. Figure 1: Some Keplerian orbits and some orbits for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Keplerian orbits with same angular momentum. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Levine’s property of Keplerian orbits with two common points. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ). The hodographs of the parabolic and hyperbolic orbits are circular arcs which end when the velocity vector is tangent to the circle. To see this, just simplify the expression (V ) of the velocity x˙ = − m C y r − β  , y˙ = m C x r − α  . (W) On the right-hand sides, the first terms form a vector of norm m|C| −1 , the second terms a constant vector [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An orbit for ρ 4 = x 4 + y 4 and its hodograph [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An orbit for ρ = 2r + y 3/r2 and its hodograph 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We present an elementary deduction of the Newtonian force from Kepler's laws. We relate it to a generalization by Jacobi of the Keplerian motion, where the Euclidean form in the plane is replaced by some function with the same homogeneity. We show how several convexity properties of the generalized Keplerian orbits appear in this context. We describe the generalized hodographs.

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 / 1 minor

Summary. The manuscript presents an elementary deduction of the Newtonian inverse-square force law from Kepler's laws of planetary motion. It connects this to a Jacobi-style generalization in which the Euclidean quadratic form is replaced by an arbitrary function of identical homogeneity degree, and examines the resulting convexity properties of generalized orbits along with their hodographs.

Significance. If the derivation and generalization hold, the result would indicate that the Newtonian force law and associated orbital properties depend on homogeneity rather than the specific Euclidean metric, offering a foundational perspective on classical mechanics with potential for non-Euclidean extensions. The elementary character of the deduction is a clear strength.

major comments (1)
  1. [§3] §3 (Jacobi generalization): the claim that the area law, hodograph closure, and convexity properties survive replacement of the Euclidean form by a generic homogeneous function of the same degree is load-bearing for the central thesis but lacks explicit verification that the second-derivative structure and radial-angular relations used in the force extraction remain intact. Homogeneity alone does not automatically preserve these features, as noted in the stress-test concern; a concrete check or counterexample for a non-quadratic homogeneous function is required.
minor comments (1)
  1. [Abstract] Abstract, paragraph 2: the phrasing 'some function with the same homogeneity' would benefit from specifying the homogeneity degree explicitly (degree 2) to avoid ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the single major comment below and will incorporate the suggested verification into a revised manuscript.

read point-by-point responses

  1. Referee: §3 (Jacobi generalization): the claim that the area law, hodograph closure, and convexity properties survive replacement of the Euclidean form by a generic homogeneous function of the same degree is load-bearing for the central thesis but lacks explicit verification that the second-derivative structure and radial-angular relations used in the force extraction remain intact. Homogeneity alone does not automatically preserve these features, as noted in the stress-test concern; a concrete check or counterexample for a non-quadratic homogeneous function is required.

    Authors: We agree that an explicit verification strengthens the central claim. The derivations in §3 rely on the homogeneity of degree 2 together with Euler’s theorem to relate the radial and angular components and to extract the force law; these steps are metric-independent once homogeneity is granted. Nevertheless, to address the concern directly we will add a concrete example in the revised §3 (or a short appendix) using a smooth, non-quadratic homogeneous function of degree 2, for instance F(x,y) = (x² + y²)·g(θ) where g is a positive periodic function that is not constant. For this choice we will recompute the second-derivative structure, confirm that the area law and hodograph closure persist, and verify the convexity properties of the resulting orbits. This explicit check will demonstrate that the features survive beyond the quadratic case. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reduction of outputs to fitted inputs or self-citations

full rationale

The paper presents an elementary geometric deduction of the inverse-square force from Kepler's laws, then relates the construction to Jacobi's historical generalization in which the Euclidean quadratic form is replaced by any homogeneous function of the same degree. The abstract and available excerpts contain no fitted parameters, no self-citations that carry the central claim, and no re-labeling of a known empirical pattern as a new prediction. The convexity and hodograph properties are derived directly from the homogeneity assumption and the area law without circular re-use of the target force law. Because the derivation remains independent of any author-specific prior constants or ansatzes smuggled through citation, the chain does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on the mathematical property that the replacement function must share the same homogeneity degree as the Euclidean distance; no free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Any function with the same homogeneity degree as Euclidean distance can replace it while preserving the dynamical structure of Kepler motion.
    Invoked when the paper relates the Newtonian deduction to the Jacobi generalization (abstract, sentence 2).

pith-pipeline@v0.9.0 · 5559 in / 1204 out tokens · 39130 ms · 2026-05-19T13:17:26.211689+00:00 · methodology

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.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We assume that ρ is differentiable enough and positively homogeneous of degree 1... There is a function ρ{2} ... ∂²ρ = ρ{2} [[y², -xy], [-xy, x²]] ... ¨x = -C²/p x ρ{2}

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

15 extracted references · 15 canonical work pages

  1. [1]

    Albouy, There is a projective dynamics , Eur

    A. Albouy, There is a projective dynamics , Eur. Math. Soc. Newsletter, 89 (2013) pp. 37–43

    work page 2013

  2. [2]

    Albouy, T

    A. Albouy, T. Stuchi, Generalizing the classical fixed-centres problem in a non Hamilto- nian way, J. Phys. A: Math. Gen., 37 (2004) pp. 9109–9123

    work page 2004

  3. [3]

    Albouy, A.J

    A. Albouy, A.J. Ure˜ na, An antimaximum principle for periodic solutions of a forced oscillator, Communications in Contemporary Mathematics, 25 (2023) 2250041

    work page 2023

  4. [4]

    Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht (1974)

    L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht (1974)

    work page 1974

  5. [5]

    Darboux, Sur une loi particuli` ere de la force signal´ ee par Jacobi, Note 11 au cours de m´ ecanique de T

    G. Darboux, Sur une loi particuli` ere de la force signal´ ee par Jacobi, Note 11 au cours de m´ ecanique de T. Despeyrous, tome 1er, A. Hermann, Paris (1884)

    work page

  6. [6]

    Gauss, Theoria Motus Corporum Coelestium in sectionibus conicis solem ambien- tium, Perthes & Besser, Hamburg (1809)

    K.F. Gauss, Theoria Motus Corporum Coelestium in sectionibus conicis solem ambien- tium, Perthes & Besser, Hamburg (1809)

    work page

  7. [7]

    Godal, Conditions of Compatibility of Terminal Positions and Velocities , Xlth Inter- national Astronautical Congress, Springer-Verlag (1961) pp

    T. Godal, Conditions of Compatibility of Terminal Positions and Velocities , Xlth Inter- national Astronautical Congress, Springer-Verlag (1961) pp. 40–44

    work page 1961

  8. [8]

    Graves, On the Motion of a Point upon the Surface of a Sphere , Proceedings of the Royal Irish Academy, volume II, No

    C. Graves, On the Motion of a Point upon the Surface of a Sphere , Proceedings of the Royal Irish Academy, volume II, No. 33 (1842) pp. 207–210

    work page

  9. [9]

    Hamilton, The hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction , Proc

    W.R. Hamilton, The hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction , Proc. Roy. Irish Acad., 3 (1845–47) pp. 344–353

    work page

  10. [10]

    Hermann, Extrait d’une lettre de M

    J. Hermann, Extrait d’une lettre de M. Herman ` a M. Bernoulli, Histoire de l’Acad´ emie royale des sciences avec les memoires de math´ ematique et de physique, Paris, (1710) pp. 102–103, 519–521

    work page

  11. [11]

    Jacobi, De motu puncti singularis , Journal f¨ ur die reine und angewandte Math- ematik, 24 (1842) pp

    C.G.J. Jacobi, De motu puncti singularis , Journal f¨ ur die reine und angewandte Math- ematik, 24 (1842) pp. 5–27

    work page

  12. [12]

    J.L. Lagrange, Th´ eorie des variations s´ eculaires des ´ el´ ements des Plan` etes, premi` ere partie, Nouveaux m´ emoires de l’Acad´ emie royale des sciences et belles-lettres, 1781 (1783) pp. 199–276

    work page

  13. [13]

    Levine, A method of orbital navigation using optical sightings to unknown land- marks, AIAA Journal, 4 (1966) pp

    G.M. Levine, A method of orbital navigation using optical sightings to unknown land- marks, AIAA Journal, 4 (1966) pp. 1928–1931

    work page 1966

  14. [14]

    Newton, Philosophiæ Naturalis Principia Mathematica, London (1687) Appendix 1

    I. Newton, Philosophiæ Naturalis Principia Mathematica, London (1687) Appendix 1. Proof that Newton implies Kepler by the eccentricity vector. The classical expression of the eccentricity vector ( α, β) in the planar Kepler problem is Equation (W ) in the form α = x r − C ˙y m , β = y r + C ˙x m (E) 9 This first integral of Newton’s equation ( N) possesse...

    work page

  15. [15]

    The key ingredient is the eccentricity vector

    improved it by presenting the 3-dimensional analog in its final form in 1782 —without vector notation. The key ingredient is the eccentricity vector. Hermann wrote the expression for one of its coordinates. Lagrange wrote all 3 coordinates. Surprisingly, this vector is sometimes called the LRL vector, for Laplace, Runge and Lenz, although none of these th...

    work page