The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I’ve mentioned before that the preface of Sussman and Wisdom’s *Structure and Interpretation of Classical Mechanics* is a good starting point for thinking about these issues. As a pointed example, in this blog post I’ll look at how badly the Legendre transform is taught in standard textbooks,^{ a } and compare it to how it *could* be taught. In a subsequent post, I’ll used this as a springboard for complaining about the way we record and transmit physics knowledge.

Before we begin: turn away from the screen and see if you can remember what the Legendre transform accomplishes *mathematically* in classical mechanics.^{ b } I don’t just mean that the Legendre transform converts the Lagrangian into the Hamiltonian and vice versa, but rather: what key mathematical/geometric property does the Legendre transform have, compared to the cornucopia of other function transforms, that allows it to connect these two conceptually distinct formulations of mechanics?… [continue reading]