A list of mechanics books (with comments,
The two traditional books for mathematicians have been:
R. Abraham and J. Marsden:
Foundations of Mechanics, Perseus publishing, second revised
edition l994, ($92 list )
This, according to one of the reviews lcarried by Amazon.com is not only a course in mechanics, but also a course in global anaylsis. It is defiitely for more sophisticated readers It also seems to be about the most expensive mechanics book at this time.
V.I. Arnold, Mathematical
Methods of Classical Physics, Graduate
Texts in Mathematics, Springer-Verlag
Most students find this much more user friendly. I don't like the fact that you start with symplectic manifolds and differential forms before you learn much
about Hamiltonian Mechanics. It is best to learn this in R 2n and then learn that everything you just did works locally on a symplectic manifold....of maybe that Hamiltonian mechanics when posed in R 2n changes coordinates locally as if what you have is a cotangent bundle, etc. Of course, this complaint is even worse in Abraham and Marsden.
A newer book that is not so well known:
J. Marsden and T. Ratiu,
Introduction to Mechanics and Symmetry,
Springer-Verlag ($64 list).
I like the classic problems that this book starts out with. I also like that Hamiltonian mechanics is in a linear space the first time around, and then one graduates to manifolds. Also, if you look carefully, the description of Lagrangian mechanics makes clear that it is coordinate invariant, and is very suitable for constrained problems. Dan thinks the notation is clumsy....with the effect that even very simple proofs are confusing and long.
Now for other possibilities:
A First Course in Mechanics,
Oxford Science Publications ($32)
This is at the level of calculus. It has big print, and nice drawings. Much of the descriptions are NOT given in the manner a modern geometer would give them. On the other hand, I think it quite possible this would be useful for developing some physical intuition (Not many mathematical sources try).
Symmetry in Mechanics, Birkhouser,
I sat through a good part of the short course that this book grew out of ( I was the organizer). The idea is excellent. Learn about all the basic ideas of mechanics and symmetry and get an introduction to both Lie groups and symplectic geometry through the central force and/or two body problem. Lots of students absolutely love this book. I like the more classical physics myself... which probably proves nothing more than that I am old fashioned? I like coordinates.
Herbert Goldstein, et al:
Hall ($146 list)
This was the classics physics text for years and years. It might be useful in getting into the minds of physicists. Note that it has been around for so many years and used in so many classes that you can practically find a free one....just look in the attic.
Landau and Lifschitz,
This is even older than Goldstein, and possibly more to a mathematicians taste. I don't own this, but I cut my teeth on The Classical Theory of Fields.
Amazon.com is offering a special on the two of them, I notice.
V. Guillemin and S. Sternberg,
Symplectic Techniques in Physics,
Cambridge ($47 list ;paperback)
I really like this book. I will use it to describe the Toda lattice, and I often refer to it. However, it sat on my bookshelf for years before I dared open it, and even longer before I understood anything at all in it. I suspect it is not an easy beginning text. Note that this does contain some stuff on stationary phase. Also, I use the end a lot more than the beginning, which I use less than the middle.