Making Sense of the Legendre Transform

TL;DR

This paper clarifies the mathematical structure and physical motivation of the Legendre transform, emphasizing its symmetry and natural emergence in thermodynamics and statistical mechanics, to improve understanding in physics education.

Contribution

It presents a modified, more symmetric presentation of the Legendre transform, connecting its mathematical properties with physical applications in thermodynamics and statistical mechanics.

Findings

Manifestly symmetric equations for the Legendre transform

Clearer geometric and algebraic understanding of the transform

Natural emergence of the transform in thermodynamics and statistical mechanics

Abstract

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.