Noether's theorem for the conditional principle of least action

TL;DR

This paper extends Noether's theorem to systems with constrained actions, linking symmetries and conservation laws in the presence of differential equation constraints without auxiliary fields.

Contribution

It introduces a novel extension of Noether's theorem for conditional extrema, emphasizing gauge symmetries of constraints and constructing conservation laws solely from original fields.

Findings

Extended Noether's theorem for constrained actions.

Constructed conservation laws from conditional symmetries.

Demonstrated the method with multiple examples.

Abstract

We consider the problem of a conditional extremum of an action in a class of fields constrained by differential equations. For this setup, we propose an extension of Noether's first theorem to connect the symmetries of the action and the imposed equations to the currents conserved at the conditional extrema. The key ingredient of the extension is the gauge symmetry of the differential equations constraining the admissible class of field configurations. We consider a special type of global symmetries of the action which we call conditional symmetries. Such global symmetries must be special cases of gauge transformations of the constraint equations. We construct conservation laws that follow from the conditional symmetries of action. No Lagrange multipliers or other auxiliary fields are introduced and the conserved currents include only the original fields. We also prove the converse theorem which connects the conserved currents to the conditional symmetries of action. The general method is illustrated by several examples.