Journal
AMERICAN JOURNAL OF PHYSICS
Volume 77, Issue 7, Pages 614-622Publisher
AMER ASSOC PHYSICS TEACHERS AMER INST PHYSICS
DOI: 10.1119/1.3119512
Keywords
educational courses; Fourier transforms; free energy; Legendre polynomials; physics education; statistical mechanics
Funding
- U. S. National Science Foundation [DMR-0705152, DUE0524987]
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The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the 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. We then discuss examples 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.
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