A Huygens-Leibniz-Lange framework for classical mechanics

J.W. van Holten

REVIEW 3 major objections 4 minor 29 references

A Huygens-Leibniz-Lange framework for classical mechanics

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Classical mechanics can be rebuilt without the primitive concept of force: three conservation-based laws reproduce the standard theory.

desk verdict A historically literate, clean axiomatic repackaging of conservative point-mass mechanics — not new physics, but a solid pedagogical core; the scope claim overreaches and the relativistic section has a genuine algebraic slip. read the letter →

arxiv 2602.14059 v1 pith:U7NMLZ6W submitted 2026-02-15 physics.hist-ph gr-qcphysics.class-ph

J.W. van Holten This is my paper

classification physics.hist-phgr-qcphysics.class-ph

keywords classicalmechanicsfoundationsofinertialframesmomentumconservationkineticenergyperpetualmotionrelativitypointparticles

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Newton's three laws are ambiguous about what force is, how inertial mass is defined, and which frames they refer to. This paper argues that all of point-mass classical mechanics can instead be derived from three laws: inertial frames exist; total momentum is a constant linear combination of velocities, defining inertial masses; and the change in kinetic energy between two configurations is path-independent, ruling out perpetual motion of the first kind. From these, force emerges as a convenient label for mass times acceleration, Newton's third law appears as momentum conservation, and interactions are captured by a potential energy depending only on relative positions. The same three principles are extended to relativistic point particles, where asymptotic four-momentum conservation plays the role of the momentum law.

What carries the argument

The machinery is the trio of laws: (1) Lange inertial frames, in which free point masses move uniformly on straight lines; (2) a constant total momentum P = Σ m_a v_a, which simultaneously defines the ratio of inertial masses; and (3) path-independence of kinetic-energy change, equivalently ∮ dT = 0 around closed configuration-space loops. The third law does the heavy lifting: it forces the existence of a potential V with ΔT = -ΔV, upgrading energy conservation and deriving the structure of interactions.

What would settle it

Observing any classical system of point masses whose kinetic-energy change around a closed path is nonzero — for example, energy radiated as electromagnetic waves during a close encounter — would violate Law 3 and show that the three laws do not cover all classical point-particle motion.

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Extended reading notes

Core claim

Newton's content is claimed to reside in three kinematic principles, not in force. First, inertial frames exist, in which free point masses move uniformly on straight lines. Second, a constant total momentum P = Σ m_a v_a exists and defines inertial masses up to a common scale. Third, kinetic-energy change between configurations is path-independent, so ∮ dT = 0 around closed loops, ruling out perpetual motion of the first kind. From these the paper derives a potential V, conservation of E = T + V, the action-reaction law, angular-momentum conservation for central forces, and interactions depending only on relative positions. The relativistic extension replaces the momentum law by asymptotic

Load-bearing premise

The load-bearing assumption is Law 3: every interaction in the isolated system is conservative, so kinetic-energy change around a closed path is zero; for real charged or gravitating point particles the mediating fields radiate energy, which the paper itself concedes, so the rederivation applies only to idealized non-radiating systems.

Editorial extensions

If this is right

Newton's second law, F = ma, becomes a naming convention rather than a physical postulate, so debates over whether it defines force or asserts something empirical lose their ground.
Momentum conservation directly yields the action-reaction content of Newton's third law and determines inertial masses from velocity changes in collisions.
The path-independence law implies that isolated point-mass interactions are conservative, with total energy T + V conserved and potentials depending only on relative positions.
Central potentials follow from angular-momentum conservation in two-body systems, giving Newtonian gravitation and Kepler's area law as corollaries.
The scheme extends to relativistic point particles, with Lorentz-invariant asymptotic four-momentum conservation replacing the classical momentum law.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the three laws are adopted as the starting point, mechanics can be introduced through energy and momentum before force, potentially reducing the conceptual gap between inertia and interaction.
The framework's own section 9 implies its strict scope: for real charged or gravitating point particles, dynamical fields carry away energy, so Law 3 and four-momentum conservation hold only ideally or asymptotically; the paper's 'all the usual results' claim should be read with that caveat.
The scalar-field example suggests a testable reinterpretation: interaction energy acts as a local shift of inertial mass (m + gφ), which could be probed by comparing inertial mass in field-free and field-rich regions.
A natural extension would be to formulate dissipative mechanics by weakening Law 3 to an inequality or by tracking energy flux into fields, connecting this axiomatics to open-system dynamics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jan-Willem has written a clear, historically literate restatement of a known set of ideas: replace Newton's force axioms by (1) inertial frames exist, (2) total momentum conservation defines mass, (3) path-independent kinetic-energy change. There is no genuinely new physics here — every ingredient is credited to Lange, Mach, Helmholtz, and Huygens — but the packing is clean and the non-relativistic core is mostly correct.

What the paper does well: it gives a coherent derivation of action-reaction from momentum conservation, shows V = V(x2 − x1) follows from translation invariance, and generalizes to N bodies. The historical discussion is careful, and the self-limitation to point masses is stated early. The relativistic section is a bonus, and the recognition that point particles radiate and therefore the four-momentum conservation is only asymptotic is honest.

The soft spots are real. First, the abstract's 'all the usual results of classical mechanics' overstates the domain. Law 3, path-independence of ΔT, is the statement that the work form is exact; for any dissipative or radiative interaction no global potential exists, and the paper itself concedes this in §9. So the framework covers conservative, non-radiating point systems — a proper subset of classical mechanics, not a replacement. Second, eq. (21) asserts the general solution of the closed-loop constraint without proof; it is a Helmholtz decomposition, and the workless a^(0) term is plausible, but it should be stated as an assumption or proved. Third, the relativistic algebra appears to have a real error: with (45), the dφ term in (46) should carry g√(1−v²), not g/√(1−v²), and the claimed conservation (47) does not follow for time-dependent fields. That error is confined to the auxiliary section, but it needs fixing before the paper is usable.

On balance: this is a competent, honest repackaging of known content. It is not a new physics result, and the scope claim should be tempered. But it is exactly the sort of paper a good pedagogical journal might want after revision. I would send it to a serious referee — the foundations are solid enough to warrant the time, and the historical attributions deserve checking.

Referee Report

3 major / 4 minor

Summary. The paper proposes an alternative axiomatic basis for classical point-mass mechanics, replacing Newton's laws by three postulates: (1) the existence of inertial frames, (2) the conservation of a linear combination of velocities (defining inertial mass and total momentum), and (3) path-independence of the change in total kinetic energy, interpreted as the impossibility of a perpetual motion of the first kind. From these postulates the author derives the existence of a potential energy and energy conservation, the action–reaction balance, angular-momentum conservation for central potentials, and an N-body generalization. A final section sketches a relativistic extension using asymptotic four-momentum conservation and a scalar mediating field. The non-relativistic core is a standard reformulation of conservative point-particle mechanics, presented in a historically informed way.

Significance. The paper has genuine pedagogical and historical value: it gathers the relevant historical threads (Huygens, Leibniz, Stevin, Mach, Lange) and shows explicitly how a force-free, momentum-and-energy based axiomatics can reproduce the standard results for conservative point-mass systems. The derivations in §§6–8 are transparent and mostly correct given the postulates. However, the central claim that this framework 'avoids the problems with Newton's formulation' and that 'all the usual results' for idealized point masses can be rederived is overstated. The third law already contains the essential conservative assumption, and the paper itself concedes in §9 that real charged or gravitating point particles violate it because they radiate. Thus the framework is an axiom system for an idealized subclass, not a complete replacement of Newtonian mechanics. With a careful revision of the scope claims and a few technical repairs, the paper could make a solid contribution to the foundations and history of classical mechanics.

major comments (3)

[§5, §9] Law 3, eq. (17), asserts path-independence of ΔT for arbitrary paths between two configurations. This is much stronger than the stated physical principle of impossibility of a perpetuum mobile of the first kind; it is essentially the assertion that all interactions are conservative (or gyroscopic). Consequently, the potential V of eqs. (21)–(24) exists only for such systems. The paper itself concedes in §9 that accelerating point particles create dynamical fields that carry away energy and that the four-momentum conservation (39) is then no longer strictly applicable. Since electromagnetic and gravitational radiation are ubiquitous, the axioms cover only a proper subclass of point-mass systems. The abstract's claim that 'all the usual results of classical mechanics, as it concerns the motion of idealized point masses, can be rederived' must be qualified to conservative, non-radiating sys
[§6, Eq. (21)] Eq. (21) is presented as 'the general solution' of the closed-loop condition (20). This needs justification. For position-dependent forces, path-independence implies that the work 1-form is exact, and the decomposition into a gradient and a zero-work term is valid. For velocity-dependent forces, the zero-work term is not necessarily of the form m a^(0) with a^(0)=v×b. More seriously, such gyroscopic terms are not automatically compatible with the second law: in a two-charge system, the magnetic forces do no work but do not conserve the sum of particle momenta m_a v_a; the missing momentum is carried by the field. Thus, unless one sets b=0 or includes field degrees of freedom, eq. (21) and the subsequent derivation of eq. (25) do not follow from the stated axioms. The manuscript should either prove the decomposition under precise assumptions or restrict the framework to forces without gyr
[§9, Eqs. (46)–(47)] The relativistic section does not derive the equations of motion from the three laws; it postulates a scalar field and its field equation. The paper states that eq. (39) is not strictly applicable when mediating fields are dynamical. In addition, the claimed equivalence in eq. (47) is not an equivalence as written: setting dE_tot/dt=0 with vanishing surface term only gives v_a · [dp_a/dt + g_a√(1-v_a^2) ∇φ(X_a)] = 0 for each particle, not the full vector equation (48). To infer the vector equation one must assume the condition holds for arbitrary initial velocities. The section should either state that additional assumption or soften the implication. As it stands, the relativistic discussion is a separate model rather than a consequence of the proposed framework, so the abstract's mention of relativistic point particles should be correspondingly demoted.

minor comments (4)

[§5, Law 3] The statement that the change in total kinetic energy depends only on the initial and final configurations C1 and C2 is ambiguous, because T is not a function of configuration alone; ΔT = T(t2)-T(t1) depends on the endpoint velocities. The law should be phrased as the path-independence of the work integral (19), with initial and final velocities specified, to avoid this imprecision.
[§1] In the quotation of Newton's third law, 'apposed' should presumably be 'opposed' (or the original spelling should be retained with an editorial note).
[§3] The text refers to 'Fig. 1' in the discussion of Stevin's proof, but no figure appears in the manuscript. Please ensure the figure is included.
[§9, after Eq. (34)] The phrase 'kinematic 4-momentum' in eq. (34) could be confused with the canonical momentum introduced later in eq. (41). Please distinguish the two notions explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three laws are openly stated postulates; energy conservation and action-reaction follow by direct unpacking, not by hidden fits or self-citation chains.

full rationale

The paper presents its own replacement laws in Section 5 as explicit postulates: the existence of inertial frames (Law 1), constancy of a chosen linear momentum combination (Law 2), and path-independence of the change in kinetic energy (Law 3, eq. 17). The later derivations are transparent mathematical consequences of these postulates. In Section 6, eq. 20 follows from Law 3 by writing dT as work, and the existence of a potential V (eq. 21) and conservation of E = T + V (eq. 24) are the standard exactness consequence of the closed-loop condition in eq. 18. This is an unpacking of the axiom, not a hidden fit: the paper never claims to derive Law 3 from weaker assumptions, nor does it fit a parameter to data and then call it a prediction. Similarly, Law 2 directly implies Δp1 = −Δp2 (eqs. 8–9), so calling this the physical content of Newton's third law is an honest equivalence, not circularity. The cited self-references ([13], [15], [25]) are historical or background remarks; the main argument is self-contained, and no load-bearing uniqueness theorem or ansatz is imported from the authors' prior work. Section 9's admission that real radiating point particles lose energy and that eq. (39) is then no longer strictly applicable is a scope limitation of the axioms, not a circular step: it narrows the domain of validity but does not make the derivation self-referential. Overall, the paper is a transparent axiomatic reconstruction rather than a circular derivation.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The three laws are the physical input; every theorem below them (F = ma as a label, action-reaction as momentum conservation, energy conservation E = T + V, N-body separation structure) is an unpacking of those axioms. No numbers are fitted to data: the inertial masses carry one global scale fixed by the choice of a mass unit (section 5), a normalization rather than a fit; the relativistic model's field mass mu and charges g are inputs from prior literature. No new entities are introduced — the scalar field is a textbook vehicle. The strongest hidden input is the conservativeness premise (law 3), which fails for radiative systems as section 9 itself concedes; eq. (21) also silently assumes simple-connectedness of configuration space.

assumptions (8)

domain assumption Existence of a Euclidean inertial frame in which all free point masses move uniformly on straight lines (1st law)
Postulated in section 5 (1st law); foundational Newtonian spacetime structure; not derived.
domain assumption Existence of a linear combination of velocities, sum m_a v_a, constant for all motions of an isolated system (2nd law)
Section 5 (2nd law); simultaneously defines inertial mass and imposes translation invariance of interactions; carries Newton's 2nd+3rd law content.
domain assumption Kinetic-energy change is path-independent: closed-loop integral of dT vanishes (3rd law)
Section 5 (3rd law); equivalent to assuming all interactions are conservative or workless; the strongest physical restriction, acknowledged in section 9 to fail for radiative systems.
domain assumption Galileo-Huygens relativity: uniform translation is indistinguishable from rest
Invoked in sections 2 and 6 to ensure no absolute frame is selected by the 1st law.
standard math Work 1-form admits Helmholtz decomposition: closed-loop work vanishing forces F = -grad V plus velocity-workless terms (eq. 21)
Section 6, eq. (21), stated as the general solution without proof; requires simply-connected configuration space, an unstated assumption.
standard math Special-relativistic kinematics p^mu = m X-dot^mu with p_mu p^mu = -m^2 c^2 (eqs. 33-35)
Section 9; standard SR background imported from prior literature.
domain assumption Asymptotic conservation of total four-momentum for locally interacting particles (eq. 39)
Section 9; assumed for the relativistic extension; the paper itself notes it fails for radiative systems.
domain assumption Minimal scalar-field model: mass replacement m + g phi and Klein-Gordon propagation (eqs. 41-44)
Section 9; illustrative toy model taken from prior literature (refs. 23-25), not derived in this paper.

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Cite this review

Pith. "Pith review of A Huygens-Leibniz-Lange framework for classical mechanics." pith.science (2026). https://pith.science/paper/U7NMLZ6W

@misc{pith2026260214059,
  author       = {Pith},
  title        = {Pith review of: A Huygens-Leibniz-Lange framework for classical mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7NMLZ6W}},
  note         = {Machine review of arXiv:2602.14059}
}

read the original abstract

I discuss the physical basis of classical mechanics, such as expressed commonly using the framework of Newton's Principia. Newton's formulation of the laws of motion is seen to have quite a few ambiguities and shortcomings. Therefore I offer an alternative set of laws, based in particular on ideas of his contemporaries Huygens and Leibniz with a crucial addition by Ludwig Lange, which avoids the problems with Newton's formulation. It is shown that from these laws of motion all the usual results of classical mechanics, as it concerns the motion of idealized point masses, can be rederived. The application of these principles to relativistic point particles is discussed.

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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