On Lagrangians with Reduced-Order Euler-Lagrange Equations

TL;DR

This paper investigates Lagrangians whose Euler-Lagrange equations have lower order than expected, revealing they must be polynomial in highest derivatives and providing explicit constructions for such cases.

Contribution

It characterizes Lagrangians with reduced-order Euler-Lagrange equations and offers a geometric construction for a family of such Lagrangians, conjecturing completeness.

Findings

Lagrangians with lower-order Euler-Lagrange equations are polynomial in highest derivatives.

Provides explicit geometric formulation for a family of such Lagrangians.

Conjectures all such Lagrangians can be obtained through this construction.

Abstract

If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such $k$-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.