On the exact inverse problem of the calculus of variations

TL;DR

This paper investigates whether a given differential equation system can be derived from a variational principle, focusing on identifying the corresponding first-order integrals for second-order Euler-Lagrange systems using an elementary approach.

Contribution

It provides a method to determine the variational integrals associated with second-order Euler-Lagrange systems, addressing the inverse problem of the calculus of variations.

Findings

Characterization of variational integrals for second-order systems

Elementary approach simplifies the inverse problem

Explicit criteria for Euler-Lagrange equivalence

Abstract

The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagrange systems.