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arxiv: 2604.24799 · v1 · submitted 2026-04-26 · ⚛️ physics.class-ph · math.HO

Recognition: unknown

The Inverse Cube Force Law

John C. Baez
Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:43 UTC · model grok-4.3

classification ⚛️ physics.class-ph math.HO
keywords inverse cube forcecentral forcesNewton's Principiaangular velocityradial motionorbital mechanics
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The pith

Newton showed that only an inverse-cube central force lets you rescale angular velocity by any constant while leaving radial motion unchanged.

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

The paper examines Newton's result from the Principia that identifies the inverse-cube law as the central force permitting a particle's angular velocity to be multiplied by a fixed factor without any change to its radial trajectory. This property sets the inverse-cube law apart from other central forces, such as the inverse-square law of gravity. The discussion covers this scaling feature together with additional dynamical properties of the same force law. A reader would care because it shows how the mathematical form of a force can separate angular and radial degrees of freedom in a precise way.

Core claim

Newton figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law. The paper discusses this and some other interesting features of the inverse cube force law.

What carries the argument

The inverse-cube central force, which alone permits uniform rescaling of angular velocity while preserving the radial equation of motion.

If this is right

  • Radial motion decouples from angular motion under this specific force law.
  • The inverse-cube law produces characteristic orbits such as Cotes spirals.
  • The same force law admits exact solutions in which angular and radial behaviors can be treated separately.
  • It supplies a concrete example of how force laws are classified by their scaling symmetries.

Where Pith is reading between the lines

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

  • The scaling property might simplify analytic or numerical treatment of problems where inverse-cube forces appear approximately.
  • It suggests a route to finding other force laws with similar separability by examining scaling symmetries in the equations of motion.
  • Modern applications could include modeling certain effective forces in atomic or plasma physics where inverse-cube terms arise.

Load-bearing premise

The force depends only on radial distance and the particle moves in a plane so that angular velocity is well-defined and can be scaled uniformly.

What would settle it

Take the equations of motion under an inverse-cube force, multiply the angular velocity by a constant factor, and check whether the radial distance as a function of time stays exactly the same.

Figures

Figures reproduced from arXiv: 2604.24799 by John C. Baez.

Figure 1
Figure 1. Figure 1: A particle spiraling into the origin in an inverse cube force. Newton’s Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently [1, 5]. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial … view at source ↗
read the original abstract

Newton's Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law! Here we discuss this and some other interesting features of the inverse cube force law.

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 / 3 minor

Summary. The manuscript claims that Newton identified in the Principia the inverse-cube central force as the unique law permitting a constant rescaling of a particle's angular velocity while leaving its radial trajectory r(t) invariant. It discusses this invariance property together with additional features of the inverse-cube force law under standard Newtonian central-force assumptions.

Significance. If the analysis is correct, the result isolates a distinctive invariance property of the inverse-cube law that is not shared by the more familiar inverse-square case, offering a clean illustration of how the centrifugal term and force law can combine to decouple angular and radial degrees of freedom. This has modest pedagogical value for classical mechanics courses and may stimulate further exploration of special central-force problems.

minor comments (3)
  1. The abstract states that the inverse-cube law 'remained little-known until fairly recently'; a citation to the modern literature that revived this result would strengthen the historical framing.
  2. The radial equation after angular-velocity rescaling is asserted to be unchanged in form; an explicit side-by-side comparison of the effective potential or acceleration term before and after rescaling would make the invariance transparent.
  3. Notation for the constant rescaling factor applied to angular velocity should be introduced once and used consistently throughout the discussion of the invariance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper discusses Newton's historical result that an inverse-cube central force permits uniform rescaling of angular velocity while leaving the radial trajectory invariant. This follows directly from the standard Newtonian central-force equations (angular-momentum conservation and the radial acceleration equation) under the assumptions of planar motion and strict radial dependence of the force. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or self-citation chain; the argument is self-contained against external benchmarks of classical mechanics and does not invoke any uniqueness theorem or ansatz from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The discussion rests on standard Newtonian central-force assumptions and historical attribution; no new free parameters, ad-hoc axioms, or invented entities are introduced in the provided abstract.

axioms (2)
  • domain assumption The force is central, depending only on the radial distance from a fixed center.
    Standard premise for all central-force problems in classical mechanics, invoked when discussing angular and radial decoupling.
  • domain assumption The motion is confined to a plane so that angular velocity is a single scalar quantity.
    Required to define a uniform rescaling of angular velocity without additional degrees of freedom.

pith-pipeline@v0.9.0 · 5351 in / 1304 out tokens · 45588 ms · 2026-05-08T04:43:19.263513+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages

  1. [1]

    S.\ Chandrasekhar, Newton's Principia for the Common Reader , Oxford U.\ Press, Oxford, 1995, pp.\ 183--200

    1995

  2. [2]

    R.\ Cotes, Harmonia Mensuarum , Cambridge, 1722

  3. [3]

    Available at https://math.ucr.edu/home/baez/physical/sarang_gopalakrishnan_thesis.pdf https://math.ucr.edu/home/baez/physical/sarang\( \; \)gopalakrishnan\( \; \)thesis.pdf

    S.\ Gopalakrishnan, Self-Adjointness and the Renormalization of Singular Potentials , B.A.\ Thesis, Amherst College, 2006. Available at https://math.ucr.edu/home/baez/physical/sarang_gopalakrishnan_thesis.pdf https://math.ucr.edu/home/baez/physical/sarang\( \; \)gopalakrishnan\( \; \)thesis.pdf

    2006

  4. [4]

    Also available at arXiv:0903.5277 http://arxiv.org/abs/0903.5277

    D.\ M.\ Gitman, I.\ V.\ Tyutin and B.\ L.\ Voronov, Self-adjoint extensions and spectral analysis in Calogero problem, J.\ Phys.\ A 43 (14) (2010), 145205. Also available at arXiv:0903.5277 http://arxiv.org/abs/0903.5277

    work page arXiv 2010

  5. [5]

    R.\ Gowing, Roger Cotes---Natural Philosopher , Cambridge U.\ Press, Cambridge, 2002

    2002

  6. [6]

    N.\ Grossman, The Sheer Joy of Celestial Mechanics , Birkh\"auser, Basel, 1996, p.\ 34

    1996

  7. [7]

    M.\ Reed and B.\ Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness , Academic Press, New York, 1975

    1975

  8. [8]

    Available at https://en.wikipedia.org/wiki/Newton's_theorem_of_revolving_orbits https://en.wikipedia.org/wiki/Newton's \;\; theorem \;\; of \;\; revolving \;\; orbits

    Newton's theorem of revolving orbits, Wikipedia. Available at https://en.wikipedia.org/wiki/Newton's_theorem_of_revolving_orbits https://en.wikipedia.org/wiki/Newton's \;\; theorem \;\; of \;\; revolving \;\; orbits