SLOW EVOLUTION IN PERTURBED HAMILTONIAN-SYSTEMS

For parametrized Hamiltonian systems with an arbitrary, finite number of degrees of freedom, it is shown that secularly stable families of e quilibrium solutions represent approximate trajectories for small (not necessarily Hamiltonian) perturbations of the original system. This b asic result is further generalized to certain conservative, but not ne cessarily Hamiltonian, systems of differential equations. It generaliz es to the conservative case a theorem due, in the dissipative case, to Tikhonov, to Gradshtein, and to Levin and Levinson. It justifies the use of physically motivated approximation procedures without invoking the method of averaging and without requiring nonresonance conditions or the integrability of the unperturbed Hamiltonian.