Noether's theorem for the conditional principle of least action

TL;DR

This work extends Noether's first theorem to systems where the action reaches extrema under differential constraints, by exploiting the gauge symmetry of the constraint equations to derive conserved currents that involve only the original fields. It introduces conditional symmetries as specializations of gauge transformations with constant parameters and proves a direct correspondence: each conditional symmetry yields a conserved current, and each current conserved on conditional extrema corresponds to a conditional symmetry. The approach avoids introducing Lagrange multipliers, addressing the issue of spurious degrees of freedom and providing explicit current formulas that respect the constrained dynamics. Through concrete mechanical and field-theoretic examples, the paper demonstrates how additional conserved quantities arise and alter the reduced dynamics, while preserving a clear Noether-type structure for conditional extremum problems.

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.