The inverse problem of the calculus of variations for systems of second-order partial differential equations in the plane

TL;DR

This paper provides a complete algebraic and differential characterization for the inverse problem of the calculus of variations applied to hyperbolic second-order PDE systems in two variables, including an algorithm and classification.

Contribution

It offers a comprehensive solution to the multiplier inverse problem for a class of hyperbolic PDE systems, including criteria, an algorithm, and classification of multiple variational formulations.

Findings

Derived necessary and sufficient conditions for variational multipliers.

Developed an algorithm for solving the inverse problem.

Classified systems with multiple variational formulations.

Abstract

A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational multiplier are derived. It is shown that the number of independent variational multipliers is determined by the nullity of a completely algebraic system of equations associated to the given system of partial differential equations. An algorithm for solving the inverse problem is demonstrated on several examples. Systems of second-order partial differential equations in two independent and dependent variables are studied and systems which have more than one variational formulation are classified up to contact equivalence.