Direct integration of generalized Lie or dynamical symmetries of three degrees of freedom nonlinear Hamiltonian systems: integrability and separability

Abstract

It is shown that by directly integrating the characteristic equation of the infinitesimals of the generalized Lie or dynamical symmetries associated with three degrees of freedom Hamiltonians, one can almost, by inspection, obtain the required involutive integrals of motion, whenever they exist. The method is illustrated for the coupled quartic and cubic oscillators considered earlier. Further, all the separable coordinates can be obtained by integrating a subset of the characteristic equation associated with the coordinate variables alone.