On the action principle for a system of differential equations

TL;DR

This paper develops methods to construct action functionals for systems of differential equations that do not naturally derive from an action principle, including conditions for existence and explicit formulations.

Contribution

It introduces criteria for the existence of a multiplier matrix to form Euler-Lagrange equations and provides explicit actions for non-variational systems, including reformulation as first-order systems.

Findings

Derived necessary and sufficient conditions for action existence.

Constructed explicit actions for systems without direct variational formulations.

Identified and described ambiguities in associating Lagrangians with differential equations.

Abstract

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler-Lagrange equations for an action principle. Two ways of action principle construction are presented. From simple consideration, we derive necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of Euler-Lagrange equations. An explicit form of the action is constructed in case if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.