Reference Request: Classical Mechanics with Symplectic Reduction

Question

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie algebra) to prove Symplectic reduction theorem (on locally free proper G-action), Arnold-Liouville Theorem (on completely integrable systems) and some more.

For instance, both Arnold's mechanics book and Spivak's physics for mathematician does not explain these concepts. I think supplements will help me understand that book's appendix (where it explains reduction theorem with lots of machinery, Ehresmann connection, and so on). Any suggestions on this?

Answer 1

As for the symplectic reduction, a good place to look at is Chapter 6 of Olver's Applications of Lie Groups to Differential Equations. This chapter is almost independent from the rest of the book.