The multiplier problem of the calculus of variations for scalar ordinary differential equations

Abstract

In the long-standing inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after being multiplied 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 $$2n, n\ge 2,$$ is proposed.