Variational principle for cylindrical curves and dynamics of spinning particles in $d=3$ Minkowski space

D. S. Kaparulin; I. A. Retuntsev; S. L. Lyakhovich

arxiv: 1907.03065 · v1 · pith:FWSJYDFWnew · submitted 2019-07-06 · ✦ hep-th · math-ph· math.MP

Variational principle for cylindrical curves and dynamics of spinning particles in d=3 Minkowski space

D. S. Kaparulin , S. L. Lyakhovich , I. A. Retuntsev This is my paper

Pith reviewed 2026-05-25 02:03 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP

keywords spinning particlesvariational principlecylindrical curvesMinkowski spacePoincare groupgauge equivalenceLagrange multipliersequations of motion

0 comments

The pith

Spinning particle dynamics in 3D Minkowski space arise from a variational principle on cylindrical curves with constant separation constraints.

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

The classical paths of massive spinning particles trace out circles on a cylinder whose axis is timelike in Minkowski space. Treating all paths on this cylinder as gauge-equivalent leads to equations of motion that are equivalent to extremizing the length of the curve while keeping the separation between points fixed. These non-Lagrangian equations are recast as an unconditional variational problem by adding the separation constraints via Lagrange multipliers. The resulting model is verified to place its states on the co-orbit of the Poincare group, reproducing the known dynamics of spinning particles.

Core claim

Starting from the observation that paths of irreducible massive spinning particles lie on a circular cylinder with timelike axis, and assuming gauge equivalence of all such paths, the equations of motion for the cylindrical curves are derived. These equations admit a variational interpretation as the conditional extremum of a length functional under constant separation conditions. Inclusion of the separation conditions via Lagrange multipliers yields an unconditional variational principle whose solutions lie on the co-orbit of the Poincare group.

What carries the argument

The length functional for curves on a cylinder, extremized subject to constant separation conditions enforced by Lagrange multipliers.

Load-bearing premise

All classical paths on the cylinder are assumed to be gauge-equivalent.

What would settle it

Computing the states from the variational principle and checking if they fail to satisfy the co-orbit condition of the Poincare group would falsify the central claim.

read the original abstract

We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we derive the equations of motion for the cylindrical curves. These equations are non-Lagrangian, but they admit interpretation in terms of the conditional extremum problem for a certain length functional in the class of paths subjected to the constant separation conditions. The unconditional variational principle is obtained after inclusion of constant separation conditions with the Lagrange multipliers into the action. We explicitly verify that the states of the obtained model lie on the co-orbit of the Poincare group. The relationship with the previously known theory is demonstrated.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete variational reformulation for 3D spinning particles by assuming all cylinder paths are gauge-equivalent, then verifies the co-orbit property, but that assumption is the weakest link.

read the letter

The core move is to start from the known cylinder geometry of massive spinning particle paths in 3D Minkowski space, assume every classical path on that cylinder is gauge-equivalent, derive the resulting non-Lagrangian curve equations, recast them as a conditional length extremum, and lift the constant-separation constraints into the action with multipliers. They then check that the solutions remain on the Poincaré co-orbit and sketch the link to earlier models. The co-orbit verification and the conditional-to-unconditional step are the parts that read as cleanly executed on their own terms. That gives the construction something definite to stand on. The assumption that all paths are gauge-equivalent is invoked without an explicit classification or proof in the abstract, and the stress-test note correctly flags it as the least-secured step. If distinct orbit classes exist that are not related by reparametrization or Poincaré transformations, the multiplier trick would not automatically reproduce the full dynamics. Nothing in the given material shows that the assumption has been independently justified or that the authors have enumerated the orbits. This is a narrow, technical piece aimed at people who already work on classical spinning particles or on variational principles in relativistic mechanics. It is incremental, but the explicit co-orbit check makes it concrete enough to be worth a referee's time rather than a desk reject. I would send it out for review so the assumption and the derivations can be examined directly.

Referee Report

2 major / 0 minor

Summary. The manuscript derives a variational principle for the worldlines of massive spinning particles in 3D Minkowski space, which are known to lie on a time-like cylinder. It assumes all classical paths on this cylinder are gauge-equivalent, obtains non-Lagrangian equations of motion for cylindrical curves, interprets them as a conditional extremum of a length functional subject to constant separation, and promotes the separation conditions to the action via Lagrange multipliers. The paper states that it explicitly verifies the resulting states lie on a Poincaré co-orbit and relates the construction to prior theories.

Significance. If the gauge-equivalence assumption is independently justified and the co-orbit verification is rigorous, the result supplies a new variational formulation for spinning-particle dynamics that could aid analysis of relativistic constraints and quantization. The explicit co-orbit check is a concrete consistency test that strengthens the claim of Poincaré invariance.

major comments (2)

[Abstract] Abstract (gauge-equivalence assumption): the derivation begins by assuming every classical path on the cylinder is gauge-equivalent under reparametrization and Poincaré action. This step is load-bearing for obtaining the non-Lagrangian EOM and for the subsequent Lagrange-multiplier construction; without an explicit orbit classification or proof that all paths (including distinct windings or separations) are connected, the claim that the final model reproduces the full co-orbit dynamics is not yet secured.
[Abstract] Abstract (verification statement): the assertion that 'the states of the obtained model lie on the co-orbit of the Poincaré group' is presented as an explicit check, yet the abstract supplies neither the explicit equations of motion nor the steps of the verification. If this check relies on the same gauge-equivalence assumption, it risks circularity and must be shown to be independent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The comments highlight important points regarding the justification of the gauge-equivalence assumption and the clarity of the co-orbit verification. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract (gauge-equivalence assumption): the derivation begins by assuming every classical path on the cylinder is gauge-equivalent under reparametrization and Poincaré action. This step is load-bearing for obtaining the non-Lagrangian EOM and for the subsequent Lagrange-multiplier construction; without an explicit orbit classification or proof that all paths (including distinct windings or separations) are connected, the claim that the final model reproduces the full co-orbit dynamics is not yet secured.

Authors: We agree that the gauge-equivalence assumption is central and would benefit from further elaboration. The assumption follows from the fact that all paths on the cylinder correspond to the same irreducible representation of the Poincaré group, with differences arising only from reparametrization and global Poincaré transformations. In the revised manuscript we will insert a short explanatory paragraph (likely in Section 2) that sketches why paths with different windings or separations remain connected under these transformations, referencing the co-orbit structure already used later in the paper. This addition will make the logical step explicit without changing the subsequent derivation. revision: partial
Referee: [Abstract] Abstract (verification statement): the assertion that 'the states of the obtained model lie on the co-orbit of the Poincaré group' is presented as an explicit check, yet the abstract supplies neither the explicit equations of motion nor the steps of the verification. If this check relies on the same gauge-equivalence assumption, it risks circularity and must be shown to be independent.

Authors: The verification is performed independently in the body of the paper: after obtaining the Euler-Lagrange equations from the unconstrained action with Lagrange multipliers, we substitute the solutions back into the Poincaré Casimir operators and confirm that the mass and spin invariants match those of the co-orbit. This calculation does not invoke the initial gauge-equivalence assumption. To remove any appearance of circularity or lack of detail, we will revise the abstract to state that the check consists of solving the derived equations of motion and verifying the Casimir invariants, and we will add a sentence in the main text explicitly noting the independence of this step from the earlier assumption. revision: partial

Circularity Check

0 steps flagged

No circularity: explicit assumption and independent verification

full rationale

The derivation begins from the stated fact that massive spinning particle paths lie on a time-like cylinder, invokes an explicit assumption of gauge equivalence among all such paths to obtain non-Lagrangian EOM, recasts them as a conditional length extremum, augments the action with Lagrange multipliers for constant separation, and then performs an explicit verification that the resulting states lie on the Poincaré co-orbit. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the gauge-equivalence assumption is presented as an input rather than derived from prior author work, and the co-orbit verification supplies independent content. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on one central domain assumption extracted from the abstract; no free parameters or new entities are mentioned.

axioms (1)

domain assumption All the classical paths on the cylinder are gauge-equivalent
This assumption is invoked to derive the equations of motion for the cylindrical curves.

pith-pipeline@v0.9.0 · 5666 in / 1108 out tokens · 30239 ms · 2026-05-25T02:03:54.016466+00:00 · methodology

discussion (0)

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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we derive the equations of motion for the cylindrical curves.

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

19 extracted references · 19 canonical work pages

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Frenkel, Z

J. Frenkel, Z. Phys. 37, 243–262 (1926)

work page 1926
[2]

Frydryszak, in: From Field Theory To Quantum Groups, p.151– 172 (World Scientiﬁc Publishing , 1996)

A. Frydryszak, in: From Field Theory To Quantum Groups, p.151– 172 (World Scientiﬁc Publishing , 1996)

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Kassandrov, N

V. Kassandrov, N. Markova, G. Schaefer, A. Wipf, J. Phys. A: Math. and Theor. 42, 315204 (2009)

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Rempel, L

T. Rempel, L. Freidel, Phys. Rev. D 95, 104014 (2017)

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Kostant, Quantization and unitary representations

B. Kostant, Quantization and unitary representations. Lectu res in modern analysis and applications III (1970) 87-208

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Souriau, Structure of dynamical systems: a symplectic view of physics

J.M. Souriau, Structure of dynamical systems: a symplectic view of physics. Vol. 149. (Springer Science and Business Media, 2012)

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[8]

Lyakhovich, A.Yu

S.L. Lyakhovich, A.Yu. Segal, and A.A. Sharapov, Phys. Rev. D 54, 5223 (1996)

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[9]

Staruszkiewicz, Acta Phys

A. Staruszkiewicz, Acta Phys. Pol. B Proc. Suppl. 1, 109–112 (2008)

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Bratek, J

L. Bratek, J. Phys. A: Math. and Theor. 43, 015208 (2010)

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Kuzenko, S.L

S.M. Kuzenko, S.L. Lyakhovich, A.Yu. Segal, A.A. Sharapov. .Int . J . Mod. Phys. A 11, 3307–3329 (1996)

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Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, J. Math. Phys . 38, 4086–4103 (1997)

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[13]

Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, Nucl. Phys. B 537, 640–652 (1999)

work page 1999
[14]

Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, Int. J. Mod.Ph ys. A 15, 4287–4299 (2000)

work page 2000
[15]

D. S. Kaparulin and S. L. Lyakhovich, Phys. Rev. D 96, 105014 (2017)

work page 2017
[16]

D. S. Kaparulin, S. L. Lyakhovich, I. A. Retuntsev, Russ. Phy s. J. 61, 2145–2154 (2019)

work page 2019
[17]

E. L. Starostin, G. H. M. van der Heijden, Monatsh. Math. 176, 481–491 (2015)

work page 2015
[18]

Gorbunov, S

I.V. Gorbunov, S. M. Kuzenko, S. L. Lyakhovich, Int. J. Mod. Phys. A 12, 4199–4215 (1997)

work page 1997
[19]

Agrachev, Y

A. Agrachev, Y. Sachkov, Encyclopaedia of Mathematical Scie nces 87 (Springer-Verlag, Berlin, Heidelberg 2004). E-mail address : dsc@phys.tsu.ru, sll@phys.tsu.ru, retuntsev.i@phys.ts u.ru Physics F aculty, Tomsk State University, Tomsk 634050, Rus sia

work page 2004

[1] [1]

Frenkel, Z

J. Frenkel, Z. Phys. 37, 243–262 (1926)

work page 1926

[2] [2]

Frydryszak, in: From Field Theory To Quantum Groups, p.151– 172 (World Scientiﬁc Publishing , 1996)

A. Frydryszak, in: From Field Theory To Quantum Groups, p.151– 172 (World Scientiﬁc Publishing , 1996)

work page 1996

[3] [3]

Kassandrov, N

V. Kassandrov, N. Markova, G. Schaefer, A. Wipf, J. Phys. A: Math. and Theor. 42, 315204 (2009)

work page 2009

[4] [4]

Rempel, L

T. Rempel, L. Freidel, Phys. Rev. D 95, 104014 (2017)

work page 2017

[5] [5]

A. A. Kirillov, Elements of the theory of group representations ( Springer-Verlag, Berlin, 1976)

work page 1976

[6] [6]

Kostant, Quantization and unitary representations

B. Kostant, Quantization and unitary representations. Lectu res in modern analysis and applications III (1970) 87-208

work page 1970

[7] [7]

Souriau, Structure of dynamical systems: a symplectic view of physics

J.M. Souriau, Structure of dynamical systems: a symplectic view of physics. Vol. 149. (Springer Science and Business Media, 2012)

work page 2012

[8] [8]

Lyakhovich, A.Yu

S.L. Lyakhovich, A.Yu. Segal, and A.A. Sharapov, Phys. Rev. D 54, 5223 (1996)

work page 1996

[9] [9]

Staruszkiewicz, Acta Phys

A. Staruszkiewicz, Acta Phys. Pol. B Proc. Suppl. 1, 109–112 (2008)

work page 2008

[10] [10]

Bratek, J

L. Bratek, J. Phys. A: Math. and Theor. 43, 015208 (2010)

work page 2010

[11] [11]

Kuzenko, S.L

S.M. Kuzenko, S.L. Lyakhovich, A.Yu. Segal, A.A. Sharapov. .Int . J . Mod. Phys. A 11, 3307–3329 (1996)

work page 1996

[12] [12]

Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, J. Math. Phys . 38, 4086–4103 (1997)

work page 1997

[13] [13]

Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, Nucl. Phys. B 537, 640–652 (1999)

work page 1999

[14] [14]

Lyakhovich, A.A

S.L. Lyakhovich, A.A. Sharapov, K.M. Shekhter, Int. J. Mod.Ph ys. A 15, 4287–4299 (2000)

work page 2000

[15] [15]

D. S. Kaparulin and S. L. Lyakhovich, Phys. Rev. D 96, 105014 (2017)

work page 2017

[16] [16]

D. S. Kaparulin, S. L. Lyakhovich, I. A. Retuntsev, Russ. Phy s. J. 61, 2145–2154 (2019)

work page 2019

[17] [17]

E. L. Starostin, G. H. M. van der Heijden, Monatsh. Math. 176, 481–491 (2015)

work page 2015

[18] [18]

Gorbunov, S

I.V. Gorbunov, S. M. Kuzenko, S. L. Lyakhovich, Int. J. Mod. Phys. A 12, 4199–4215 (1997)

work page 1997

[19] [19]

Agrachev, Y

A. Agrachev, Y. Sachkov, Encyclopaedia of Mathematical Scie nces 87 (Springer-Verlag, Berlin, Heidelberg 2004). E-mail address : dsc@phys.tsu.ru, sll@phys.tsu.ru, retuntsev.i@phys.ts u.ru Physics F aculty, Tomsk State University, Tomsk 634050, Rus sia

work page 2004