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arxiv: 1907.05667 · v1 · pith:LUJ7HJ4Mnew · submitted 2019-07-12 · 🧮 math-ph · math.MP

Tulczyjew's derivations and intrinsic field equations in classical field theories

Modesto Salgado , Silvia Vilari\~no This is my paper

Pith reviewed 2026-05-24 22:23 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Tulczyjew derivationsEuler-Lagrange equationsintrinsic formulationssingular Lagrangiansnon-holonomic constraintsclassical field theoriesvariational principles
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The pith

Tulczyjew derivations provide intrinsic Euler-Lagrange equations for singular Lagrangians in field theories

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

This paper shows how Tulczyjew's derivations from mechanics extend to classical field theories, yielding variational and intrinsic versions of the Euler-Lagrange field equations. These versions apply to the classical, implicit, and non-holonomic cases without requiring the Lagrangians to be regular. The approach is illustrated with Navier's equations and the non-holonomic Cosserat rod. It also discusses the Hamiltonian formulation when the Lagrangian is hyperregular.

Core claim

Tulczyjew's derivations can be extended to classical field theories to obtain intrinsic versions of the implicit and non-holonomic Euler-Lagrange field equations that remain valid even for singular Lagrangians, with the variational and intrinsic forms not requiring regularity of the Lagrangian functions.

What carries the argument

Extension of Tulczyjew's derivations to provide intrinsic field equations via variational principles.

If this is right

  • The classical Euler-Lagrange field equations admit intrinsic forms applicable to singular Lagrangians.
  • Implicit Euler-Lagrange field equations and their non-holonomic versions can be formulated intrinsically without regularity assumptions.
  • Navier's equations serve as an example where this method applies to a singular Lagrangian case.
  • The non-holonomic Cosserat rod provides another demonstration of the approach.
  • The Hamiltonian case is addressed when the Lagrangian is hyperregular.

Where Pith is reading between the lines

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

  • This suggests that similar extensions could handle other singular systems in continuum mechanics.
  • Connections to geometric mechanics might be explored for broader applications in field theories.
  • Further work could test the method on additional examples with non-holonomic constraints.

Load-bearing premise

The derivations from mechanics extend validly to field theories for singular Lagrangians.

What would settle it

Finding a singular Lagrangian field theory where the extended Tulczyjew intrinsic equations do not match the expected physical behavior would disprove the extension.

Figures

Figures reproduced from arXiv: 1907.05667 by Modesto Salgado, Silvia Vilari\~no.

Figure 1
Figure 1. Figure 1: Cosserat rod A Cosserat rod can be thought of as a long and thin deformable body. We assume that its length is significantly larger that its radius. A Cosserat rod can be visualized as a curve s → r(s) ∈ R 3 , [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
read the original abstract

This work presents the variational principles and the intrinsic versions of several equations in field theories, in particular, for the Classical Euler-Lagrange field equations, the implicit Euler-Lagrange field equations and the non-holonomic implicit Euler-Lagrange field equations. The advantages of the variational and intrinsic versions of these equations is that the Lagrangians functions are not necessary regular Lagrangians. We present two examples of this situation: Navier's equations and the non-holonomic Cosserat rod. Finally we comment the Hamiltonian case when the Lagrangian is a hyperregular function.

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 paper extends Tulczyjew's geometric constructions via Tulczyjew triples on jet bundles to obtain variational principles and intrinsic formulations of the classical Euler-Lagrange field equations, the implicit Euler-Lagrange field equations, and the non-holonomic implicit Euler-Lagrange field equations. It claims these formulations remain valid without requiring regular Lagrangians, supplies explicit derivations for the field-theoretic setting, and illustrates the approach with Navier's equations (treated implicitly) and the non-holonomic Cosserat rod; a brief comment addresses the Hamiltonian side restricted to the hyperregular case.

Significance. If the derivations hold, the work supplies a geometric framework that handles singular Lagrangians in classical field theory, a setting common in applications. Credit is due for the explicit field-theoretic derivations and the two concrete examples, which directly support the central claim that the constructions do not presuppose regularity.

minor comments (2)
  1. [Introduction] The introduction would benefit from an explicit statement of the main result (e.g., a numbered theorem summarizing the intrinsic non-holonomic equations) to make the logical structure clearer before the examples.
  2. Notation for the jet-bundle constructions and the Tulczyjew triple should be collected in a short preliminary section or table to aid readers unfamiliar with the mechanics case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance for singular Lagrangians in field theory, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript extends Tulczyjew's established geometric constructions (Tulczyjew triples on jet bundles) to derive intrinsic variational formulations of the implicit and non-holonomic Euler-Lagrange field equations. These derivations are presented as direct mathematical extensions without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claims to unverified inputs. The examples (Navier's equations and Cosserat rod) serve as concrete applications rather than circular validations. The Hamiltonian discussion is explicitly restricted and non-load-bearing. The work is self-contained against external benchmarks from prior independent literature, consistent with the default expectation of no circularity for such theoretical extensions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, ad-hoc axioms, or invented entities can be identified from the given text.

axioms (1)
  • standard math Standard axioms of differential geometry and variational calculus are assumed to hold for the intrinsic formulations.
    The paper invokes Tulczyjew derivations and intrinsic equations, which rest on these background mathematical structures.

pith-pipeline@v0.9.0 · 5620 in / 1176 out tokens · 51027 ms · 2026-05-24T22:23:52.907044+00:00 · methodology

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Reference graph

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