Alternative to Arnold's mathematical methods

Question

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics.

Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to be able to participate in some courses. I will attach the relevant list of subjects found at the table of contents via a picture below.

The professor incharge claimed he studied the subjects from this book and does not know of a better reference (also claims he thinks chapter 3 and 4 are very clear, which wasn't the case to me)

As such I would be happy for literature references.

What can I do as an undergrad math student?

Are there any alternatives that are more comprehensive, and less difficult to read? Mathematically rigorous as well.

Answer 1

I'm a physics professor, and more mathematically inclined than your average physicist. Arnol'd is a beautiful textbook, but it is most emphatically not the one to learn Lagrangian mechanics from. (I used it as a student when I took a graduate-level class on the material.) Here are some texts that are often used for teaching Lagrangian & Hamiltonian dynamics in upper-division undergraduate physics classes:

• Taylor's Classical Mechanics is what I've used the last few times I've taught the course to undergraduates. It's a lot clearer than a lot of texts. However, it doesn't cover some topics in very much depth. In particular, the notion of "phase space" (which is essential for understanding Arnold's methods) is given a pretty cursory treatment. It's also not always as mathematically rigorous as might be desired.
• Thornton & Marion's Classical Dynamics of Particles and Systems goes into a bit more depth on some of the topics, and is also somewhat more rigorous. However, it's a good deal more expensive than Taylor's book.
• Lanczos's Variational Principles of Mechanics was mentioned in the comments above. It's pretty good, though written in an older, dryer style, and it focuses more on the mathematical aspects of the material than the physical ones. Rigor-wise, it's a step up from Thornton & Marion. It's also dirt-cheap.
• Goldstein, Poole, & Safko's Classical Mechanics is a commonly-used graduate-level text on the subject. It's got a good degree of mathematical rigor, about the same as Lanczos; and it's somewhat more physically focused than Lanczos (as opposed to mathematically.) It was quite pricey the last time I checked.

To my recollection (though I haven't looked at Goldstein in years), none of these texts have the same level of mathematical rigor and rarefication as Arnold. If you pick one of the first two texts, there will probably be some material in Arnold that you will have not seen before.

There are multiple threads over at Physics StackExchange that might also be helpful, depending on whether you're learning the subject for the physics or for the mathematics:

• Classical Mechanics without Coordinates book
• Book about Classical Mechanics
• Which Mechanics book is the best for [a] beginner in math major?