A generalization of Noether's theorem based on the virtual work functional

TL;DR

This paper extends Noether's theorem to non-conservative systems using a virtual work functional, deriving balance principles for non-conserved currents instead of conservation laws.

Contribution

It introduces a generalized Noether's theorem based on the virtual work functional that applies to non-conservative and non-holonomic systems, expanding the scope of variational principles.

Findings

Derivation of balance principles for non-conserved currents.

Application to mechanics of points, rigid bodies, and deformable bodies.

Extension of Noether's theorem to non-exact virtual work functionals.

Abstract

In a series of previous articles by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems, as well as ones that subject to non-holonomic constraints by starting with the virtual work functional instead of an action functional. In this article, it is shown that when one starts with a virtual work functional that is not exact, so it does not admit an action functional, instead of conservation laws the extension of Noether's theorem gives balance principles for non-conserved currents. Examples of this generalized Noether identity are given in the mechanics of points, rigid bodies, and deformable bodies.