The Lazy Universe: An Introduction to the Principle of Least Action by Jennifer Coopersmith
There is an interesting phenomenon of popular versions of more technical books. We have Feynman and Hibbs’ Quantum Mechanics and Path Integrals being popularized in the form of Feynman’s QED. David Friedman turned his Price Theory textbook into the wonderful Hidden Order. And now we have an answer to the question: but who will write a “popular” (for non-expert audiences with college-level math training) version of Cornelius Lanczos’ The Variational Principles of Mechanics? Jennifer Coopersmith, that’s who.
I really enjoyed this book. The principle of least action is one of those great gems that bears returning to again and again. One of my pet pursuits is finding more and better explanations for the age-old question “Why is the classical Lagrangian kinetic minus potential energy?” (see here for example for my attempt to understand it a bit better in the case of gravity). Coopersmith devotes Section 6.6 to that very topic.
The Lazy Universe is structured by beginning with some historical material and some mathematical preliminaries (advisory note: you do need to know vector calculus to really profit from this book). Then we march from the principle of virtual work, to D’Alembert’s principle, on to Lagrangian mechanics and then Hamiltonian mechanics. Unlike Susskind’s popular classical mechanics book Coopersmith doesn’t address Poisson brackets. I really like the structure and how she builds from one chapter to the next; I certainly did not sufficiently appreciate the point of virtual work and D’Alembert when I took my classical mechanics class. In the penultimate chapter there’s a survey of how the principle of least action takes its form in other branches of physics.
I’m just not sure who this book is for, exactly. Much of the mathematical nitty-gritty and nearly all the examples are pushed to a multitude of appendices, which makes it annoying to flip back and forth if you want the full story. This was done I think so that less mathematically-adept readers could enjoy the main thread of the argument, but that still requires knowledge of the variational calculus (for example, the Euler-Lagrange equations are not derived like in Susskind’s book). So I would recommend this to people for whom this is their second (or third, etc) pass at analytical mechanics.