On the equivalence of Euler-Lagrange and Noether equations

TL;DR

This paper proves the equivalence of Euler-Lagrange and Noether equations for certain nonlinear variational problems, with implications for conservation laws and inverse problems in nonlinear PDEs.

Contribution

It establishes the conditions under which Euler-Lagrange and Noether equations are equivalent for nonlinear Poisson equations and related Lagrangians, including p-Laplacian types.

Findings

Proved equivalence of Euler-Lagrange and Noether equations under specific conditions.

Demonstrated applications to nonlinear Poisson equations and their generalizations.

Discussed inverse problems related to Lagrangian determination from symmetries.

Abstract

We prove that on the condition of non-trivial solutions, the Euler-Lagrange and Noether equations are equivalent for the variational problem of nonlinear Poisson equation and a class of more general Lagrangians, including position independent and of p-Laplacian type. As applications we prove certain propositions concerning the nonlinear Poisson equation and its generalisations, the equivalence of admissible and inner variations and discuss the inverse problem of determining the Lagrangian from conservation or symmetry laws.