Want to really understand the math approach to classical physics? This book might be the answer.

The Theoretical Minimum is a good title but its sub-title "What you need to know to start doing physics" is misleading. You can do some physics with much less and even top level physics without understanding very much in this book at all. Some of the publicity for this book also claims that it is suitable for the beginner and that it is what you need to read as a way into physics. It isn't.

This is not a book to get you started in physics and it isn't a book that you need to read to do physics.

This mis-targeting by the publisher is one reason why some readers might really hate this book. Its clear message about some very messy areas of classical physics is one good reason why some readers will love it. However, because it has some chapters included to make it look like a book for beginners some of the readers who would benefit from reading might well be put off.

The most important thing to realize is that this is two books, by two very different authors, pushed together. There is a deep thinking account of the modern approach to classical mechanics and a very simple introduction to calculus and Newtons laws of motion.  The problem is that no one reader is going to want to read both.

Lecture 1 is an interesting and simple account of the notion of state space, determinism and reversibility. It ends with Interlude 1, which is basic revision on vectors and trig. Lecture 2 continues with a very basic look at motion, or rather a basic introduction to differential calculus. It does get as far as simple harmonic motion, but there are lots of accounts of this sort that do the job just as well and at the greater length that the true beginner requires. Interlude 2 is about integral calculus. Lecture 3 continues the simple stuff with a look at dynamics and perfectly standard study of Newtonian mechanics followed up by Interlude 3, which introduces partial differentiation. 

Lecture 4 expands mechanics into a system of particles and introduces ideas like phase space and conserved quantities. This is where the nature of the book starts to change. From here the concerns become increasing sophisticated. Lecture 5 is about energy and you need to read this one carefully because among all of the standard things you know about energy there are some interesting comments. 

Lecture 6 is where the account really takes off - no more simple F=ma but full on Lagrangian mechanics. The idea of the principle of least action is introduced together with the Euler-Lagrange equations as the basic equations of motion. It also attempts to make the reader really understand why the Lagrangian is a better approach to mechanics than the more direct and more obvious F=ma.

This is the chapter where the book gets difficult. If the reader has only just been hanging on this is the chapter that sees them fall unless they put a lot of work in. The level of mathematical sophistication required is quite high, despite the fact that everything is explained in a plain speaking a language as possible. 

Lecture 7 is about symmetry and contains one of the nicest and simplest explanations of Noether's theorem you can hope for. Chapter 8 makes the connection with the Hamiltonian formulation of mechanics which is so important for non-relativistic quantum mechanics. This is the chapter were you first realise that even though QM isn't a topic of this book - there is a follow up book on it - it is behind the scenes most of the time. The book is attempting to give you the basics of classical mechanics formulated in a way that makes the transition to QM seem more reasonable. By the end of lecture 8 you should have both the Lagrangian formulation and the Hamiltonian formulation under your belt and you should begin to see why each has its place. 

Lecture 9 moves on to some of the high level theorems and advantages you get from such a sophisticated view point. We first have the Gibbs- Liouville theorem and then in the next lecture the Poisson Bracket or PB. If you have struggled with this particular concept before you will most likely be suddenly handed an epiphany - "Ah so this is what the PB is all about".

The final lecture is a lightening (pun intended) tour of electromagnetism which is where classical mechanics really gets interesting. The book closes with an appendix on the classical problem of central forces and planetary orbits. This is a key analysis simply because so many systems turn out to be things revolving round other things. 

This is a great book if you have encountered enough physics to know that you don't quite get Lagrangian mechanics. The reason is simple enough because after reading it you will get it and Hamiltonian mechanics and a lot of other sophisticated ideas. If you are such a reader then I would suggest reading Lecture 1 carefully, then skim read the next two and their associated interludes and restart at Lecture 4. Whatever you do don't give up on the book because of the pages devoted to "what is a derivative". 

What if you actually need to know what a derivative is? 

My advice would be to pick up another book. This one is altogether too short and gets too deep to provide a route into physics from that starting point. 

The bottom line is that this is a highly recommended book if you have enough physics and or enough math. 

I look out for the review of the latest volume in the series - Special Relativity and Classical Field Theory: The Theoretical Minimum,

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