Noether’s theorem for the conditional principle of least action

arXiv:2602.06059v1 Announce Type: new
Abstract: We propose an extension of the first Noether’s theorem to the theories whose field equations result from the requirement of conditional extremum of the action. 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. We also prove the converse theorem which connects the conserved currents to the conditional symmetries of action. The general method is illustrated by several mechanical and field-theoretical examples.