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arxiv: 2409.11063 · v3 · submitted 2024-09-17 · ⚛️ physics.class-ph · cs.RO· math-ph· math.MP

Variational approach to nonholonomic and inequality-constrained mechanics

A. Rothkopf , W. A. Horowitz This is my paper

Pith reviewed 2026-05-23 20:34 UTC · model grok-4.3

classification ⚛️ physics.class-ph cs.ROmath-phmath.MP
keywords nonholonomic mechanicsvariational principlesLagrange-d'Alembert equationsconstrained dynamicsSchwinger-Keldysh formalismaction extremizationinequality constraints
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The pith

Nonholonomic systems admit an explicit scalar action whose extremization recovers the Lagrange-d'Alembert equations.

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

The paper constructs a general action functional for mechanical systems that obey non-integrable velocity constraints or position inequalities. This action is obtained by taking the classical limit of the Schwinger-Keldysh formalism. Extremizing the action directly produces the correct equations of motion for such systems. Validation occurs through numerical optimization on standard examples, which reproduces known trajectories without first writing differential equations. The result supplies a variational route to dynamics that previously lacked a general action principle.

Core claim

We construct an explicit and general action for non-holonomic motion, motivated by the classical limit of the quantum Schwinger-Keldysh action formalism. Our formulation recovers the correct dynamics of the Lagrange-d'Alembert equations via extremization of a scalar action. We validate the approach on canonical examples using direct numerical optimization of the novel action, bypassing equations of motion. Our framework extends the reach of variational mechanics and offers new analytical and computational tools for constrained systems.

What carries the argument

The scalar action obtained from the classical limit of the Schwinger-Keldysh formalism, whose stationary paths enforce the nonholonomic constraints.

If this is right

  • Direct numerical optimization of the action reproduces the trajectories of constrained systems without deriving differential equations first.
  • The same construction applies to both non-integrable velocity constraints and inequality constraints on position.
  • Conserved quantities remain accessible through the symmetries of the new action.
  • The method supplies a variational starting point for analytical approximations in constrained mechanics.

Where Pith is reading between the lines

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

  • The approach could be tested on time-dependent constraints or systems with friction to check whether the action still yields consistent results.
  • It may allow path-integral style calculations for classical nonholonomic problems that are currently handled only by differential-equation methods.
  • Numerical codes that optimize actions could be repurposed for nonholonomic examples, potentially improving stability in long-time simulations.

Load-bearing premise

The classical limit of the Schwinger-Keldysh action formalism produces a variational principle whose stationary paths satisfy the nonholonomic Lagrange-d'Alembert equations.

What would settle it

Numerical minimization of the proposed action for a vertical rolling disk on an inclined plane should produce the same path and rotation history as integration of the corresponding Lagrange-d'Alembert equations.

Figures

Figures reproduced from arXiv: 2409.11063 by A. Rothkopf, W. A. Horowitz.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Variational principles play a central role in classical mechanics, providing compact formulations of dynamics and direct access to conserved quantities. While holonomic systems admit well-known action formulations, non-holonomic systems -- subject to non-integrable velocity constraints or position inequality constraints -- have long resisted a general extremized action treatment. In this work, we construct an explicit and general action for non-holonomic motion, motivated by the classical limit of the quantum Schwinger-Keldysh action formalism, rediscovered by Galley. Our formulation recovers the correct dynamics of the Lagrange-d'Alembert equations via extremization of a scalar action. We validate the approach on canonical examples using direct numerical optimization of the novel action, bypassing equations of motion. Our framework extends the reach of variational mechanics and offers new analytical and computational tools for constrained systems.

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 / 2 minor

Summary. The manuscript constructs an explicit scalar action for nonholonomic and inequality-constrained mechanical systems, motivated by the classical limit of the Schwinger-Keldysh formalism. It claims that stationary paths of this action recover the Lagrange-d'Alembert dynamics, and validates the claim through direct numerical optimization of the action on canonical examples, bypassing explicit solution of the equations of motion.

Significance. If the central construction holds, the work supplies a long-sought general variational principle for systems with non-integrable velocity constraints and position inequalities. The explicit action and the numerical optimization route (which supplies independent confirmation that extremization yields the correct trajectories) are notable strengths; they open routes to both analytic conserved quantities and computational methods that avoid deriving the constrained equations first.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'rediscovered by Galley' appears without a citation; add the reference at first mention in the abstract or introduction.
  2. [Numerical validation] The numerical examples are described only at the level of 'canonical examples'; a short table or paragraph listing the specific systems (e.g., rolling disk, skate on ice) and the observed error metrics would strengthen the validation section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, including the recognition of the explicit action construction and the numerical validation approach. The recommendation for minor revision is appreciated. No specific major comments were provided in the report, so we have no points requiring detailed rebuttal or revision at this stage.

Circularity Check

0 steps flagged

Derivation from external quantum formalism with independent numerical validation

full rationale

The paper motivates its action via the classical limit of the Schwinger-Keldysh formalism (an external reference, Galley), then validates by direct numerical optimization on canonical examples that recovers Lagrange-d'Alembert dynamics without using the equations of motion. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim has independent content from the external motivation and numerical bypass.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5673 in / 1021 out tokens · 49305 ms · 2026-05-23T20:34:49.975950+00:00 · methodology

discussion (0)

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Reference graph

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