There aren't a lot of up-to-date treatments of the calculus of variations for mathematicians nowadays,let alone physics students. To be fair,though,that's changing recently with the publications of the texts by van Brunt and Dracogna. Both those treatments are rather analytical and probably not what you're looking for.
The best and most complete introduction to the subject I've seen is in the last 4 chapters of S.P.Novikov and I.P Tiamanov's Modern Geometric Structures and Fields,available through the AMS.It walks the reader through not only modern formulations of classical and relativistic mechanics on both semi-Reimannian and symplectic manifolds with variational methods,the last chapter gives an in-depth derivation of the Yang-Mills equations using these methods.
I think you may find everything you're looking for in this book.
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