Noether theorem for mu-symmetries

TL;DR

This paper extends Noether's theorem to mu-symmetries, linking them to modified conservation laws and exploring their relation to standard symmetries and invariants in variational problems.

Contribution

It introduces a version of Noether's theorem for mu-symmetries, connecting these symmetries to modified conservation laws and variational principles.

Findings

Mu-symmetries lead to mu-conservation laws, sometimes reducing to standard conservation laws.

Mu-symmetries relate to conditional invariants.

Modified Euler-Lagrange equations arise under mu-symmetries.

Abstract

We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting mu-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.