Legendre transform

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,I was pleased to note as this essay went to press that my choice of Landau, Goldstein, and Arnold were confirmed as the “standard” suggestions by the top Google results.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.If not, can you remember the definition? I couldn’t, a month ago.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? (Analogously, the question “What is useful about the Fourier transform for understanding translationally invariant systems?” can be answered by something like “Translationally invariant operations in the spatial domain correspond to multiplication in the Fourier domain” or “The Fourier transform is a change of basis, within the vector space of functions, using translationally invariant basis elements, i.e., the Fourier modes”.) The status quo Let’s turn to the canonical text by Goldstein for an example of how the Legendre transform is usually introduced. After a passable explanation of why one might want to move from a second-order equation of variables to a first-order equation of variables, we hear this [3rd edition, page 335]: Treated strictly as a mathematical problem, the transition from Lagrangian to Hamiltonian formulation corresponds to the changing of variables in our mechanical functions from to by [ ]. The procedure for switching variables in this manner is provided by the Legendre transformation, which is tailored for just this type of change of variable. Consider a function of only two variables , so that a differential of has the form (8.3)   where (8.4)   We wish now to change the basis of description from to a new distinct set of variables , so that differential quantities are expressed in terms of differentials and Let be a function of and defined by the equation (8.5)   A differential of is then given as     or, by (8.3), as     which is exactly in the form desired. The quantities and are now functions of the variables and given by the relations (8.6)   which are analogues of Eqs. (8.4). The Legendre transformation so defined is used frequently in thermodynamics. The first law of thermodynamics… Huh? Did you see an clean definition of a function transform in there? is supposed to be a function of and , but the right-hand side of (8.5) has dependence. Can we always find a way to eliminate for arbitrary ? What does it mean when we can’t, or there are multiple solutions? And in what sense can become a variable independent of if its definition, , depends on ? Contrast this to the Fourier, special conformal, or Laplace transforms, which are unambiguous ways to convert a function of one variable to a function of another. If you can reconstruct a clean definition using this quote, it will be ugly and you will do so by implicitly drawing on your previously obtained knowledge of when one can and cannot treat variables as independent (knowledge that is not accessible to the student reader) and by making assumptions that are true for physical Lagrangians but not true generally (surprise! has to be convex in ). And the motivation for the definition — beyond merely “look at how pretty Hamilton’s equations turn out to be” — will still be opaque. It would be bad enough if this was just Goldstein because that book is, to my knowledge, the most widely used mechanics textbook, presumably representing the level of clarity achieved by the modal physicist. But I sat down in the library where the classical mechanics books are kept and flipped through seven or eight moreLandau & Lifshitz, Hand & Finch, Rossberg, plus a bunch I hadn’t heard of before.c   and they were as bad or worse. The venerable textbook by Landau, for instance, uses the same ambiguous differential notation and declines to explain what the Legendre transform is in general; rather, it just declares a formula for the Hamiltonian in terms of the Lagrangian [Vol. 1, 3rd edition, page 131]: (40.2)   Notice how the functional parameters are written for the Hamiltonian but not the Lagrangian? It is an impressive sleight of hand designed to distract you from the weird fact that this definition implicitly requires inverting the equation by solving for in terms of , , and , and then inserting back into Eq. (40.2). Indeed, serious ambiguities arise when you start trying to literally interpret a quantity with differentials in terms of different variables, some of which are independent and some of which are not. (Remember, when starting from a Lagrangian defined on space, is generically a function of both and .To keep my blood pressure in check, I’m just going to skip completely over the fact that this Hamiltonian has dependence. Almost all physics textbook fail to clearly explain to the student why we can nevertheless indulge in the sin of pretending that is an independent variable from , attributable perhaps to the mystery of faith. For clarity on this, I suggest picking up Gelfand and Fomin’s “Calculus of variations”, which is well regarded… <a class="continue-reading-link" href="https://blog.jessriedel.com/2017/06/28/legendre-transform/">Continue reading</a>