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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?

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I think you should go for the canonical reference Foundations of mechanics by Abraham and Marsden. It has pretty much everything that you are talking about. –  Sandeep Thilakan Feb 2 at 7:59
    
    
Arnold-Liouville theorem is in Arnold's book. –  MBN Mar 24 at 7:35
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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.

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