Prerequisites for Arnold's Methods of Classical Mechanics

I've finished with Gregory's classical mechanics and was looking for something a bit more challenging. I thought Arnold's methods of classical mechanics look pretty interesting, but it's definitely more mathematically complex than anything I would have done before, especially the bits about manifolds and differential forms - which I know essentially nothing about.

Do you think I'd be able to get anything out of it, or is the mathematical background required just too immense for now? Some other good choices might be Sommerfeld, Landau or Goldstein, which are maybe more physicsy. I wondered if anyone had some good advice... thanks!

Arnold's classical mechanics is very mathematical, but a great introduction to the subjects you mention. I think you can never go wrong with Sommerfeld and Landau/Lifshitz. Goldstein has to be read with a grain of salt, particularly concerning the issue of nonholonomous constraints (if I remember right, the treatment using d'Alembert's principle is correct but the treatment with Hamilton's principle is wrong; it's correct in Landau+Lifshitz vol. 1).

Arnold is great if you want an introduction to differential geometry, But I recommend first an intermediate course in mechanics.

Looking at gregory's book's table of content, I think canonical transformation and the Hamilton-Jacobi theory are missing.

So, if you are really (really) interested in the specific differential geometry stuff applied to mechanics, go for Arnold. Otherwise, differential geometry methods can wait until you know more stuff about analytical mechanics.

Awesome, thanks. I think Landau has a section on the canonical equations, so in that case I'll attempt Landau before Arnold. That will probably take a little while, but should be good preparation

18-years-old are you already know the word "manifold"... wow.

Btw, being an engineer with 0 background in the subject I highly recommend Lanczos "The variational principles of mechanics". Pretty underrated book in my opinion, but one of the best I've ever read.

Landau I don't like because it does not indulge much in explanations: you understand ? good. you don't ? good. But it is a great book nonetheless. Lanczos on the other hand spends lots of time explaining even"trivial" stuff and I particularly enjoy the way he writes. Plus it is a very cheap book (Dover) which is always a pro. It also stresses the importance between differential geometry and classical mechanics several times throughout the book, but without getting too technical.

PS: probably I find Landau too hard because of my background, but you seem far more skilled than I am so you won't have any problem.

dRic2 said:

you already know the word "manifold"

I know the name... but I wouldn't say I know what it means, or any of the mathematics that describes them

dRic2 said:

Btw, being an engineer with 0 background in the subject I highly recommend Lanczos "The variational principles of mechanics". Pretty underrated book in my opinion, but one of the best I've ever read.

Landau I don't like because it does not indulge much in explanations: you understand ? good. you don't ? good. But it is a great book nonetheless. Lanczos on the other hand spends lots of time explaining even"trivial" stuff and I particularly enjoy the way he writes. Plus it is a very cheap book (Dover) which is always a pro. It also stress the importance between differential geometry and classical mechanics several times throughout the book, but without getting too technical.

I did see that book recommended somewhere! I already started Landau yesterday so I think I'm going to try and stick with that, but I can pick up a copy of Lanczos probably next week to see if I like that also. Thanks!

etotheipi said:

I know the name... but I wouldn't say I know what it means, or any of the mathematics that describes them

Still impressive. At that age I wasn't even aware of the existence of integrals...

etotheipi said:

I did see that book recommended somewhere!

Probably it was me answering an other "book to learn analytical mechanics"-type post... I just copy-paste the same answer every now and then

I read some of Arnold when I started learning symplectic/contact geometry to get some physical intuition for the subject. It was definitely useful for that purpose, but maybe the fact that math people like the book isn't always the greatest advertisement for a physics text...

Infrared said:

I read some of Arnold when I started learning symplectic/contact geometry to get some physical intuition for the subject. It was definitely useful for that purpose, but maybe the fact that math people like the book isn't always the greatest advertisement for a physics text...

I would say it's actually a good book for physicists, I didn't find it as daunting as I thought. Unlike Abraham and Marsden, that book still give me nightmares.

Arnold's Mathematical Methods of Classical Mechanics is very good book and introduction, worth to look at is also Ralph Abraham Classical Mechanics but all these books make sense with connection with Henri Cartan , H.Flanders - differential forms.

Infrared said:

I read some of Arnold when I started learning symplectic/contact geometry to get some physical intuition for the subject. It was definitely useful for that purpose, but maybe the fact that math people like the book isn't always the greatest advertisement for a physics text...

May be it's true. Spivak is very mathematical.

Arnold really can be read in your first year of university. It would be a mistake to avoid this book, alongside other less sophisticated treatments of mechanics.

I have been asked on some occasions to say something about differential forms. A symbol like '⨛f' is the 'same' but 'more advanced' as '∫f'

⨛f being the upper Riemann integral