Conservation laws for under determined systems of differential equations

TL;DR

This paper generalizes conservation laws to under determined systems of differential equations by extending Ibragimov's theorem, defining adjoint equations and formal Lagrangians, and establishing a Noether theorem for such systems.

Contribution

It introduces a framework for deriving conservation laws for under determined systems, expanding the applicability of Ibragimov's conservation theorem.

Findings

System and its adjoint form a variational problem.

Inherited symmetries lead to conservation laws.

A Noether theorem is established for these systems.

Abstract

This work extends the Ibragimov's conservation theorem for partial differential equations [{\it J. Math. Anal. Appl. 333 (2007 311-328}] to under determined systems of differential equations. The concepts of adjoint equation and formal Lagrangian for a system of differential equations whose the number of equations is equal to or lower than the number of dependent variables are defined. It is proved that the system given by an equation and its adjoint is associated with a variational problem (with or without classical Lagrangian) and inherits all Lie-point and generalized symmetries from the original equation. Accordingly, a Noether theorem for conservation laws can be formulated.