Null Lagrangians in Schwarzian mechanics

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.