Can you suggest some good sites for studying Lagrangian and Hamiltonian dynamics and approximation methods in?

Answer 1

No good sites I know, but there are books:
Classical Mechanics by Landau and Lifshitz
Classical Mechanics by Goldstein
Also (depending on exactly what approach you want to research) books by Rabinovich sometimes co-authorred by Trubetskov
Libraries would have them

There is some basic stuff here:
http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_1/node4.html