A Note on the Legendre Transformation

TL;DR

This paper offers a geometric perspective on the Legendre transformation using Legendrian lifts, explores its properties and singularities, and discusses applications to Clairaut equations and contact transformations.

Contribution

It introduces a geometric approach to the Legendre transformation via Legendrian lifts and examines its properties, singularities, and applications to differential equations and contact transformations.

Findings

Geometric understanding of Legendre transformation

Explanation of singularities in dual curves

Application to Clairaut differential equations

Abstract

We present the Legendre transformation in a geometric way based on the procedure of the Legendrian lift. This approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities of dual curves. Also we consider application of the Legendre transformation to the Clairaut differential equation. Finally, we say a few words class of contact transformations and present an infinite group of contact transformations different from the Legendre transformation. Keywords: Legendre transformation, duality, contact transformation, contact structure, pedal curve, singular point, Clairaut equation