Classical mechanics has always been a source of interesting dynamics problems. Many ideas in dynamical systems theory (and the rest of mathematics, for that matter) were first developed in an attempt to understand the Newtonian n-body problem or the motion of a rigid body. I will try to present the theory in a modern, geometrical style and illustrate it with many concrete examples. Topics will include as many of the following as time permits:
Newtonian and Lagrangian mechanics, calculus of variations Variational existence proofs Hamiltonian and Poisson mechanics, symplectic structures Symmetry, Noether's theorem, reduction, Liouville's theorem Integrable systems Non-integrable systems, perturbation theory, Birkhoff normal form, applications of KAM theory
Basic theory of ordinary differential equations, manifolds, differential forms.
None required, but here are two recommended books.
"Mathematical Methods of Classical Mechanics (2nd edition)", by V.I. Arnold. This is one of the classic mathematics texts in the subject; a source of inspiration for anyone interested in mechanics even if it leaves some of the details to the reader.
"Notes on Dynamical Systems", by Moser and Zehnder
"Introduction to Hamiltonian Dynamics and the N-body Problem", by Meyer, Hall and Offin "Foundations of Mechanics", by Abraham and Marsden "Introduction to Mechanics and Symmetry", by Marsden and Ratiu "Mathemtical Aspects of Classical and Celestial Mechanics", by Arnold, Koslov and Neistadt "Lectures on Celestial Mechanics", by Siegel and Moser