Direct integration of generalized Lie symmetries of nonlinear Hamiltonian systems with two degrees of freedom: integrability and separability

Abstract

Many of the integrable coupled nonlinear oscillator systems are associated with generalized Lie symmetries involving velocity dependent terms. For a class of systems with two degrees of freedom, the authors show that by integrating the characteristic equation associated with the generalized symmetries, the required involutive integrals of motion can be obtained explicitly in a straightforward manner, almost by inspection and without recourse to Noether's theorem. Further, all the separable coordinates can be obtained by integrating a subset of the characteristic equation associated with the coordinate variables alone. The explicit examples include the two coupled generalized Henon-Heiles, quartic, sextic and other polynomial oscillator systems as well as the perturbed Kepler system.