Averaging method for quasi-integrable Hamiltonian systems

A deterministic averaging method for quasi-integrable Hamiltonian systems is proposed to predict the approximate response of many degree-of-freedom autonomous or non-autonomous strongly nonlinear systems. The form and dimension of the averaged equations depend on the number of degree-of-freedom and the number of resonant relations of the associated Hamiltonian systems. In non-resonant case, the averaged equations for n action variables or n independent integrals of motion are derived. In resonant case, the averaged equations for n action variables and α combinations of angle variables are derived. Two examples are given to illustrate the application of the proposed method. It is shown that the analytical results agree well with those from numerical solution even for systems with very strong nonlinearity.