Making Sense of the Legendre Transform
R. K. P. Zia (Virginia Tech), Edward F. Redish (U Maryland), and Susan R. McKay (U. Maine)
The Legendre Transform (LT) is a common feature of many upper division and graduate physics classes. However, discussions of it tend to be ad hoc, poorly motivated, and confusing. As a result, the LT equations become something to be memorized without understanding. In this paper we describe a more satisfying way of looking at LT relations both mathematically and physically. Mathematically this results in highly symmetric equations that clarify the structure of the transform both algebraically and geometrically. Physically, we motivate the transform as an issue of choosing independent variables that are easily controlled and give examples drawn from classical mechanics and thermodynamics. In thermodynamics, we demonstrate how the LT arising naturally from statistical mechanics and show how use of dimensionless thermodynamic potentials lead to more natural and symmetric relations.
COMMENTARY: This paper isn't really about PER, but the writers are fascinating bunch. Joe Redish is, well, Joe Redish. Susan McKay is founding director of the University of Maine Center for Science and Mathematics Education Research (where I work, after all...). And together with Zia, a friend, they wrote a paper which gets deeply into the meaningfulness of some mathematics used in physics. Papers like these might serve as a push for certain kinds of research on student understanding at the upper level, but that's not why I'm posting. I'm posting because 2 of the authors are mentors of mine, so there!