Hamiltonian and non-Hamiltonian perturbation theory for nearly periodic motion

Kruskal’s asymptotic theory of nearly period motion [M. Kruskal, J. Math. Phys. 4, 806 (1962)] (with applications to nonlinear oscillators, guiding center motion, etc.) is generalized and modified. A new more natural recursive formula, with considerable advantages in applications, determining the averaging transformations and the drift equations is derived. Also almost quasiperiodic motion is considered. For a Hamiltonian system, a manifestly Hamiltonian extension of Kruskal’s theory is given by means of the phase-space Lagrangian formulation of Hamiltonian mechanics. By performing an averaging transformation on the phase-space Lagrangian for the system (L → L̄) and adding a total derivative dS/dτ, a nonoscillatory Lagrangian Λ=L̄+dS/dτ is obtained. The drift equations and the adiabatic invariant are now obtained from Λ. By truncating Λ to some finite order in the small parameter ε, manifestly Hamiltonian approximating systems are obtained. The utility of the method for treating the guiding-center motion is demonstrated in a separate paper.