Derivation of conservation laws from nonlocal symmetries of differential equations

Abstract

An identity is derived which yields a correspondence between symmetries and conservation laws for self-adjoint differential equations. This identity does not rely on use of a Lagrangian as needed to obtain conservation laws by Noether’s theorem. Moreover, unlike Noether’s theorem, which can only generate conservation laws from local symmetries, the derived identity generates conservation laws from nonlocal as well as local symmetries. It is explicitly shown how Noether’s theorem is extended by the identity. Conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds. The constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws.