At Least Three Solutions for the Hamiltonian System and Reduction Method

We investigate the multiplicity of solutions for the Hamiltonian system with some asymptotically linear conditions. We get a theorem which shows the existence of at least three periodic solutions for the asymptotically linear Hamiltonian system. We obtain this result by the variational reduction method which reduce the infinite dimensional problem to the finite dimensional one. We also use the critical point theory and the variational method.{\bf Key Words and Phrases:} Hamiltonian system, asymptotical linearity, variational reduction method, critical point theory, variational method, (P.S.) condition..