Misc,
Making Sense of the Legendre Transform
(2008)cite arxiv:0806.1147Comment: 11 pages, 3 figures.
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.
