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I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. Perhaps something a bit more recent. There are a few things I would like it to have.
At least as much differential geometry as Arnold. Even better would be a dedicated section on DG at the start to cover everything needed to read the book.
It must use local coordinate systems and not try to do everything in a coordinate free way.
The book Foundations of Mechanics by Abraham and Marsden doesn't use local coordinates (and focuses on Hamiltonian mechanics more so than Lagrangian). The book by Spivak is nice too and but there's something about it that doesn't quite suit me.