Null Lagrangians in Schwarzian mechanics

TL;DR

This paper explores null Lagrangians in Schwarzian mechanics, revealing their properties, relation to energy functions, and introducing higher-order null Lagrangians using Schwarzian derivatives.

Contribution

It introduces higher-order null Lagrangians in Schwarzian mechanics based on Schwarzian derivatives, expanding the understanding of null Lagrangians beyond standard cases.

Findings

Null Lagrangians satisfy Euler-Lagrange equations and are total derivatives.

Null Lagrangians are characterized by vanishing energy functions.

Higher-order null Lagrangians are constructed using Schwarzian derivatives.

Abstract

In addition to standard and non-standard Lagrangians of classical mechanics, we consider, in this work, null Lagrangians that (i) identically satisfy the Euler-Lagrange equation and at the same time can be expressed as (ii) the total derivative of some scalar function. As an addendum to the properties in (i) and (ii) we find that null Lagrangians are also characterized by (iii) vanishing energy functions or Jacobi integrals. By working with higher-order SL(2;R) invariant Schwarzian derivatives introduced recently by Krivonos we demonstrate that these Schwarzians, especially the even-order ones, provide a natural basis to introduce higher-order null Lagrangians in Schwarzian mechanics.