Can a student with a heavy math background start learning physics with Goldstein's "Classical Mechanics"?
Or is the book too obtuse with basic physics that I need to start elsewhere?
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You can, certainly, it is a good book. But there are many other good books that you might also appreciate, with more advanced math. E.g. the "classics"
V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Vol. 60, Springer, 1989
R. Abraham and J.E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 2008.
J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999.
Michael Spivak's new book on mechanics (Spivak, Physics for Mathematicians. Mechanics I, 2010) seems to be great, too. See http://mathpop.com/mechanics1.htm.