On the inverse problem of calculus of variations

TL;DR

This paper investigates conditions under which a Lagrangian can be derived for fourth-order ordinary differential equations, focusing on cases where the third derivative is absent, and employs the Jacobi last multiplier and symmetries.

Contribution

It extends inverse calculus of variations methods to fourth-order ODEs with specific conditions, linking Jacobi last multiplier and Lie symmetries for Lagrangian determination.

Findings

Lagrangians can be derived for certain fourth-order ODEs with absent third derivatives.

The Jacobi last multiplier is effectively used to find Lagrangians in these cases.

Application to physics equations demonstrates the method's practical utility.

Abstract

We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348 (1996) 5007-5029] which ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. The key is the Jacobi last multiplier as in the case of a second-order equation. Two equations from a Number Theory paper by Hall, one of second and one of fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally the Lagrangian of two fourth-order equations drawn from Physics are determined with the same method.