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

## TL;DR

This paper develops variational and intrinsic formulations of classical field equations, including Euler-Lagrange and non-holonomic equations, allowing analysis without regular Lagrangians, with applications to Navier's equations and Cosserat rods.

## Contribution

It introduces intrinsic variational principles for various field equations that do not require regular Lagrangians, extending the applicability of these formulations.

## Key findings

- Intrinsic formulations work for non-regular Lagrangians.
- Examples include Navier's equations and Cosserat rod.
- Discussion of Hamiltonian case with hyperregular Lagrangian.

## 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.