Transition to the case of "resolved gauge" in the Lagrange-Poincar\'e equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

S. N. Storchak

arxiv: 1906.08561 · v1 · pith:XDNBTDCAnew · submitted 2019-06-20 · 🧮 math-ph · math.MP

Transition to the case of "resolved gauge" in the Lagrange-Poincar\'e equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

S. N. Storchak This is my paper

Pith reviewed 2026-05-25 19:15 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Lagrange-Poincaré equationsprincipal fiber bundlemechanical system with symmetryresolved gaugeindependent coordinatesscalar particlesRiemannian manifoldLie group action

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The pith

Assuming parametric representations for local sections exist in the principal bundle, the Lagrange-Poincaré equations transition to independent coordinates.

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

The paper continues earlier derivations by taking Lagrange-Poincaré equations written in dependent coordinates for a mechanical system with symmetry and rewriting them in independent coordinates. The system models two interacting scalar particles on a Riemannian manifold formed as the product of a principal fiber bundle total space and a vector space, with free proper isometric action by a compact semisimple Lie group. The transition rests on the existence of parametric representations for local sections. A sympathetic reader would care because the change produces the resolved-gauge form of the equations, removing coordinate dependencies that complicate the description of the motion.

Core claim

Assuming the existence of the parametric representations for local sections in the principal bundle, we make the transition to independent coordinates in the obtained Lagrange-Poincaré equations.

What carries the argument

Parametric representations for local sections in the principal fiber bundle, which convert the dependent-coordinate Lagrange-Poincaré equations into independent-coordinate form for the resolved gauge.

If this is right

The equations now use only independent coordinates in the resolved gauge case.
The description applies directly to the two-particle system on the given Riemannian manifold.
The compact semisimple Lie group symmetry remains encoded in the new equations.
The base of the principal bundle is the bundle space of the associated bundle.

Where Pith is reading between the lines

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

The same transition technique could apply to other mechanical systems reduced by symmetry.
Independent coordinates may simplify finding first integrals or equilibrium solutions.
Numerical integration of the dynamics would operate with fewer variables after the change.
The approach might extend to cases with different group actions or non-Riemannian metrics.

Load-bearing premise

The existence of parametric representations for local sections in the principal bundle.

What would settle it

An explicit principal bundle and local section for which no parametric representation can be written, so that the coordinate transition cannot be carried out.

read the original abstract

This note is a continuation of our earlier articles arXiv:1612.08897 and arXiv:1709.09030, where using the dependent coordinates the local Lagrange-Poincar\'e equations were obtained for a mechanical system with symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and vector space), on which a free proper and isometric action of a compact semisimple Lie group is given. Assuming the existence of the parametric representations for local sections in the principal bundle, we make the transition to independent coordinates in the obtained Lagrange-Poincar\'e equations.

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

This is a narrow technical continuation that rewrites prior Lagrange-Poincaré equations in independent coordinates under one added assumption.

read the letter

This note continues Storchak's earlier preprints on the Lagrange-Poincaré equations for two interacting scalar particles on a product manifold with a compact semisimple Lie group action. It takes the dependent-coordinate form from those papers and, assuming parametric representations for local sections exist, rewrites the equations using independent coordinates on the bundle space. The transition itself is presented as a direct re-expression once the assumption is granted. If someone is already working inside that exact setup, the independent form removes the extra variables and could simplify later calculations. The paper is explicit about the assumption and stays within the framework of the prior work, so the algebra appears consistent on its own terms. The main limitation is that the result inherits its content almost entirely from the two cited preprints; the new step is the coordinate change itself rather than any broader method or application. The assumption about parametric representations is stated but neither constructed nor verified here, which leaves the equations conditional. No general theorems, no new reductions, and no engagement with wider questions in geometric mechanics appear. The citation pattern is almost entirely self-referential. This is aimed at the small set of readers already following this author's sequence on reduced equations for this specific symmetric system. Most people working in mathematical physics or mechanics will need the earlier papers first and will still face the unexamined assumption. I would not bring the paper to a reading group or cite it. It does not seem substantial or independent enough to justify sending out for peer review.

Referee Report

1 major / 2 minor

Summary. This short note continues the author's prior works (arXiv:1612.08897, arXiv:1709.09030) on Lagrange-Poincaré equations in dependent coordinates for a mechanical system with symmetry: two interacting scalar particles on the product of a principal fiber bundle total space and a vector space, equipped with a free, proper, isometric action of a compact semisimple Lie group. Under the explicit assumption that parametric representations exist for local sections of the principal bundle, the note performs the change to independent coordinates in the previously derived equations.

Significance. If the coordinate transition is valid under the stated assumption, the result supplies a more conventional independent-coordinate form of the reduced equations, which may facilitate concrete calculations or comparisons with other geometric-mechanics treatments. The work's value is incremental and tied to the two earlier preprints; no new physical predictions, machine-checked proofs, or parameter-free derivations are presented here.

major comments (1)

[Abstract] Abstract and opening paragraph: the central claim (transition to independent coordinates) is conditioned on the existence of parametric representations for local sections, yet the note provides no verification or construction of these representations for the specific manifold under consideration. This assumption is load-bearing; without it the transition cannot be performed.

minor comments (2)

The title is excessively long and contains quotation marks around a phrase; a shorter, more conventional title would improve readability.
The manuscript refers to equations from the two earlier arXiv preprints without restating their numbering or key forms; adding a brief self-contained recap of the dependent-coordinate equations would make the transition step easier to follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading of our short note. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and opening paragraph: the central claim (transition to independent coordinates) is conditioned on the existence of parametric representations for local sections, yet the note provides no verification or construction of these representations for the specific manifold under consideration. This assumption is load-bearing; without it the transition cannot be performed.

Authors: We agree that the assumption of the existence of parametric representations for local sections is load-bearing for the coordinate transition performed in the note. The manuscript is a brief continuation of our prior works and therefore states the assumption without supplying an explicit construction or verification for the specific manifold (product of the principal-bundle total space with a vector space). We will revise the abstract and opening paragraph to make this reliance explicit and to add a short clarifying remark that such parametric representations are guaranteed locally by the definition of a principal bundle (local triviality) together with the choice of coordinates on the base manifold; we will also note that the concrete parametrization depends on the particular bundle and is not constructed here. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a short continuation note that takes the Lagrange-Poincaré equations already derived in dependent coordinates from the author's prior arXiv preprints as its starting point and performs an explicit coordinate transition under the stated assumption that parametric representations for local sections exist. This transition is presented as a distinct calculational step rather than a re-expression that reduces the claimed result to the inputs by construction. Self-citations are present but supply only the input equations; the new content is the change to independent coordinates, which does not rely on an unverified uniqueness theorem or ansatz imported from the same author chain. No load-bearing circular step is exhibited by the paper's own text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard differential geometry (principal bundles, Lie group actions, Riemannian metrics) plus the two prior papers; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence of parametric representations for local sections in the principal bundle
Invoked in the abstract as the assumption that enables the transition to independent coordinates.
domain assumption The manifold is a product of the total space of a principal fiber bundle and a vector space with a free proper isometric action of a compact semisimple Lie group
Stated as the setting for the mechanical system in the abstract.

pith-pipeline@v0.9.0 · 5662 in / 1273 out tokens · 21151 ms · 2026-05-25T19:15:42.302659+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

S. N. Storchak. The Lagrange-Poincar´ e equations for a mech anical sys- tem with symmetry on the principal ﬁber bundle over the base repre - sented by the bundle space of the associated bundle. arXiv:1612.08 897 [math-ph]

work page
[2]

S. N. Storchak. Coordinate representation of the Lagrange- Poincar´ e equations for a mechanical system with symmetry on the total spa ce of a principal ﬁber bundle whose base is the bundle space of the asso ci- ated bundle. arXiv:1709.09030 [math-ph] 11

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

S. N. Storchak. The Lagrange-Poincar´ e equations for a mech anical sys- tem with symmetry on the principal ﬁber bundle over the base repre - sented by the bundle space of the associated bundle. arXiv:1612.08 897 [math-ph]

work page

[2] [2]

S. N. Storchak. Coordinate representation of the Lagrange- Poincar´ e equations for a mechanical system with symmetry on the total spa ce of a principal ﬁber bundle whose base is the bundle space of the asso ci- ated bundle. arXiv:1709.09030 [math-ph] 11

work page internal anchor Pith review Pith/arXiv arXiv