The Schwarzian derivative and Euler--Lagrange equations

TL;DR

This paper explores the Schwarzian derivative's role in variational calculus, revealing it as both a first integral and an Euler--Lagrange operator for specific Lagrangians, thus linking it to fundamental variational principles.

Contribution

It demonstrates that the Schwarzian derivative can be interpreted as a first integral and an Euler--Lagrange operator, providing new insights into its variational properties.

Findings

Schwarzian derivative is a first integral of a second order Lagrangian's Euler--Lagrange equation.

It acts as the Euler--Lagrange operator for a specific class of variations.

The study connects the Schwarzian derivative to variational calculus and differential equations.

Abstract

We study the Schwarzian derivative from a variational viewpoint. Firstly we show that the Schwarzian derivative defines a first integral of the Euler--Lagrange equation of a second order Lagrangian. Secondly, we show that the Schwarzian derivative itself is the Euler--Lagrange operator for an appropriately chosen class of variations.