The search for periodic solutions of Hamiltonian equations originated  in celestial mechanics. It has given rise to many developments in  mathematics. The Poincare - Birkhoff fixed point theorem, for example,  inspired V. I. Arnold to formulate his conjectures about forced 
oscillations of Hamiltonian vector fields. Solutions of these conjectures led A. Floer to the Floer homology theories, a main tool in symplectic topology. In order to establish periodic solutions on a  given energy surface, Paul Rabinowitz invented in the 1980-ties  powerful variational technics. They allowed the construction of  symplectic invariants which relate periodic orbits of Hamiltonian  Systems to symplectic rigidity phenomena. More recently, the A.Weinstein conjecture about periodic orbits on contact type energy  surfaces was studied by H. Hofer and others using PDE. methods of  pseudoholomorphic curves. Nowadays these methods play an important  role in the symplectic field theory, which, however is still under construction.