The Multiplier Problem of the Calculus of Variations for Scalar Ordinary Differential Equations

TL;DR

This paper addresses the inverse calculus of variations problem for scalar ODEs of order 2n, proposing a method to find a Lagrangian and multiplier that transform the equation into an Euler-Lagrange form.

Contribution

It introduces a new approach to solve the multiplier problem specifically for scalar ordinary differential equations of order 2n, expanding the understanding of inverse variational problems.

Findings

Provides a solution method for the multiplier problem in scalar ODEs of order 2n.

Establishes conditions under which a differential equation can be derived from a variational principle.

Enhances the theoretical framework for inverse problems in the calculus of variations.

Abstract

In the inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after multiplying with the multiplier, becomes the Euler--Lagrange equation for the Lagrangian. An answer to this problem for the case of a scalar ordinary differential equation of order $2n, n\geq 2,$ is proposed.