Question

I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think of the fields that arise in physics as sections of vector bundles (or maybe principal bundles) and would love an approach to classical mechanics or what have you that took advantage of this.

Now for the questions:

Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?
What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)
Are there benefits to thinking about things this way, ie would it be of benefit to someone attempting to learn this material to do it this way?

Answer 1

1.

I am in love with Fecko's Differential Geometry and Lie Groups for Physicists. Despite not being just about mechanics (but rather about more or less all rudimentary modern theoretical physics) it discusses both Lagrangian and Hamiltonian formalism. It also provides countless exercises (with nice hints) so that you can really get a feel for the matter.
2.

I can't think of any major drawbacks. Of course, if the problem has no symmetry you sometimes have no other choice than to go back to some coordinates and solve numerically. But this is probably non-issue for you because I suppose you first want to understand physical problems with some structure.
3.

There are countless benefits. To list just few of them.

relation to symmetries and conserved quantities becomes obvious. Noether's theorem in Hamiltonian formalism is so amazingly simple statement (Hamiltonian is constant for symmetry flow if an only if the generator of the symmetry is constant for Hamiltonian flow) that one has to wonder where all the long-winded coordinate calculations went.

Not only are the calculations short, one also gains valuable geometrical insights e.g. about the flow of the configuration on the manifold.

It's a beautiful formalism.

I don't know about others but whenever I have to calculate in coordinates I become nervous. I can compute the results but after few pages when most of the quantities mysteriously cancel, you don't really know why what you derived is true. So then you go back to geometry and lo and behold, the derivation is just few lines and obvious. Of course I am exaggerating now but that's the way I feel.

It's the basis for all of modern physics. If the above four points were true in classical mechanics, they are even more true when dealing with things like gauge theories (and that is where the full beauty and power of mathematics comes out)