Analytical Mechanics is a very powerful tool used mostly in Physics (which I study), but also Applied Mathematics and some Engineering branches.
Long story short, it's a formulation of Physics in terms of a more rigorous and very general mathematical formalism, which allows you to deal in a relatively simple way with very complicated problems.
There are two main formalisms in Analytical Mechanics: one is due to Luigi Lagrange (hence "lagrangian mechanics") and the other to William Hamilton (hence "hamiltonian mechanics").
They describe the same Physics but differ in some key aspects which make them more suitable for specific kind of problems: the lagrangian formalism works fine with Classical Mechanics' problems, but with Quantum Mechanics' problems the hamiltonian formalism is far more used. The lagrangian formalism comes back in some applications of Quantum Mechanics like Path Integral, some Quantum Electrodynamics and Quantum Field Theory.
I've enjoyed studying both of them.
Self shot - March 2012.