I am looking for a book that axiomatizes classical (meaning Newtonian) mechanics, as some books do with special relativity. I would be very interested in reading such a book.
Although its notation is kind of outdated, I find Foundations of Mechanics by Abrahams and Marsden a very nice book that really axiomatizes Classical Mechanics. I should warn you, however, that the Math requirements are beyond simple-minded calculus, it requires you to be familiar with Elementary Differential Geometry. The first chapter of the book covers the necessary mathematical background wonderfully, assuming you already mastered Calculus, undergraduate Linear Algebra, and undergraduate Real Analysis.
It is a great book for the ones who like to be inbetween Math and Physics. Someone clearly more oriented towards Physics may find the book very rigid.
As other people mentioned, Lagrangian and Hamiltonian mechanics are the modern versions of mechanics. If you are comfortable with Goldstein, then you can start reading V. I. ARNOLD - CLASSICAL MECHANICS or JOSE, SALETAN- CLASSICAL MECHANICS Roughly, thep aximatization of Lagrangian leads to rigorous calculus of variation (it is interpreted as functional derivative of lagrangian ) And aximatization of Hamiltonian mechanics leads to study of Symplectic manifold. Hope this will give a good start.