The author wrote the book based on the more advanced writings of Cornelius Lanczos, and you can think of it as a "popularization", but it is more than just a simple retelling. It is a history and a philosophy of advanced mechanics. It starts off with a fun look at the early days of Galileo, the Bernoullis, and so on. This is the part of the book that is most enjoyable; partly because the history is interesting and partly because of the engaging style. However, it might not be a style everyone likes. It is slightly academic and tends towards footnotes and references rather than keeping the flow going. There are also far too many appendices that give examples and expand on what is being explained. The body of the book is less than 100 pages, but there are over 200 pages of appendices.

Chapter 4 starts us on a quest to understand where the principle of least action comes from. First we look at the principle of virtual work. A strange idea when you first meet it, and to be honest no less strange after its explanation. Then on to D'Alembert's principle, which is slightly less strange for seeming so obvious.

From this point the book starts to be less rewarding. The problem is that in the early chapters it promises so much - an explanation of least action as a shortest path in some sort of abstract space, but we never really get the explanation. Instead Chapter 6 discusses Lagrangian mechanics, including the less well known version with Lagrange multipliers. Then we have an explanation of Hamiltonian mechanics. The problem is that these explanations leave a lot to be figured out. As the subject matter gets harder, the explanations become more vague and increasingly reference other material.

The book misses many opportunities to address issues that really throw the beginner and also fails to point out many things that would make things simpler. For example, the well known issue of why you can treat position and velocity as independent variables when they are clearly related is only touched on briefly. For me it is the key to understanding that the Euler Lagrange equations give you the equations of motion which supplies the connection between position and velocity.

Despite all my reservations this is an excellent book and I enjoyed reading it, even though it didn't deliver on its potential to make me see the unifying principles that make variational calculus so natural in a wide range of physical phenomena. It is clear that it is often something to do with finding the geodesic in some abstract space, but this generalization isn't precise.

I recommend this book if only because it will make you think harder about the way it all works. It doesn't give you the entire picture, but what it does give you is well worth it.

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