We consider Arnoldi like processes to obtain symplectic subspaces for Hamiltonian systems. Large systems are locally approximated by ones living in low dimensional subspaces; we especially consider Krylov subspaces and some extensions. This will be utilized in two ways: solve numerically local small dimensional systems or in a given numerical, e.g. exponential, integrator, use the subspace for approximations of necessary functions. In the former case one can expect an excellent energy preservation. For the latter this is so for linear systems. For some second order exponential integrators we consider these two approaches are shown to be equivalent. In numerical experiments with nonlinear Hamiltonian problems their behaviour seems promising.