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What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model? Thanks for ideas.

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Besides the answers given you could look at Dubrovin, Fomenko and Novikov's Modern Geometry GTM series – Steve Huntsman Jul 29 '10 at 1:53
VERY good suggestion,Steve-the new epic by Novikov and Tiamanov I referenced below by no means supercedes this classic. – The Mathemagician Jul 29 '10 at 3:47
thanks, I will look at those references – user4 Aug 22 '10 at 21:20

4 Answers 4

up vote 3 down vote accepted

I think that the book

  • David Bleecker: Gauge Theory and Variational Principles, Addison-Wesley, 1981

contains exactly what you are looking for.

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You might want to have a look at my book, "The Geometrization of Physics" which is freely available online in numerous places, in particular from or directly from my homepage:

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this is very useful – user4 Aug 22 '10 at 21:21

The section on classsical field theory in IAS QFT Volume 1 is a pretty good place to get started.

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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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