I am a math grad student who would like to learn some classical mechanics. The caveat is I am not to interested in the standard coordinate approach. I can't help but think of the fields that arise in physics are actually 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?
thanks for your time, sean