Which is the best book in modern theoretical classical mechanics?

[camel_jockey]

Hi!

I am a very mathematically-oriented physicist. Since I never plan in making contact with "dirty" mechanics like robotics, structural problems or force diagrams, I want a book that prepares me for the mathematical/theoretical foundations of mechanics so that I can transition more smoothly to string theory and quantum field theory - where the action, the Hamilton-Jacobi equations and symmetries/Noether currents are in focus. Is there any book which "has it all" ?

I was wondering if someone could recommend me a CONCISE, preferably short, book which fulfils this in the language of differential geometry etc. An internet PDF would also be of interest...

Many thanks!

 

[deluks917]

Mathematical Methods of Classical Mechanics by Arnold seems like a good choice for you.

The following webpage seems like it might be very usefull (it has two set of course notes):

http://math.ucr.edu/home/baez/classical/

 

[lithis]

https://www.amazon.com/dp/0914098322/?tag=pfamazon01-20 on Amazon is enlightening, especially

It is quite clear that differential geometry is assumed. (Well, Spivak suggests that the first two volumes of "A Comprehensive Introduction to Differential Geometry" should be read before hand.)

There is a thorough discussion of Lagrangian and Hamiltonian mechanics from the differential geometric perspective.

and

There's an entire chapter (26 pages) dedicated to the Hamilton-Jacobi theory.

You can see the http://olivier.thill.perso.neuf.fr/books/bospphma.htm" online, where several pages can be previewed as well.

http://www.math.uga.edu/~shifrin/Spivak_physics.pdf" is a 100-page PDF for some lectures Spivak gave; it is based on the first part of this book.

 

[camel_jockey]

Thanks! I am checking out the PDF now and will see if I have cash for Spivaks monster book!

 

[rezeew]

I cannot recommend one specific "best" book in modern theoretical classical mechanics as it ultimately depends on the individual's learning style and background knowledge. However, I can provide some suggestions for books that cover the mathematical and theoretical foundations of mechanics and prepare readers for more advanced topics like string theory and quantum field theory.

One book that comes to mind is "Classical Mechanics: The Theoretical Minimum" by Leonard Susskind and George Hrabovsky. This book covers the fundamentals of classical mechanics in a concise and rigorous manner, using the language of differential geometry and symplectic geometry. It also includes a chapter on Hamiltonian mechanics and Noether's theorem, which are important concepts in modern theoretical mechanics.

Another book that may be of interest is "Classical Mechanics" by Herbert Goldstein. This book is more comprehensive and covers a wider range of topics in classical mechanics, including Lagrangian and Hamiltonian formalism, rigid body dynamics, and small oscillations. It also has a section on canonical transformations, which are relevant for understanding symmetries in mechanics.

For a more advanced and mathematical approach, "Mathematical Methods of Classical Mechanics" by V.I. Arnold is a highly recommended text. It covers the mathematical foundations of classical mechanics, including symplectic geometry, Hamiltonian dynamics, and variational principles. It also includes a chapter on the mathematical aspects of Noether's theorem.

As for a concise and freely available resource, "Lectures on Classical Mechanics" by John C. Baez is a great option. It covers the basics of classical mechanics in a clear and concise manner, using a geometric approach. It also has a chapter on symmetries and Noether's theorem.

Ultimately, it is important to find a book that aligns with your learning style and background knowledge. I suggest exploring these options and seeing which one resonates with you the most. Good luck in your search!