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.

Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains

“…To derive these symmetries we make use of the contact transformations (vide Eqs. (18) and (25)) obtained earlier.…”

“…We first identify the contact transformation between the variables of the nonlinear ODE and the associated linear ODE (vide Eq. (18) or (25)). Using this transformation we can express the partial derivatives that appear in the vector field (29) in terms of the variables of the nonlinear ODE and then compare the resultant expression with the vector field (31) which in turn provides the dynamical symmetries for the nonlinear ODE.…”

“…From the contact transformation (for example Eq. (18) or (25)), we can extract the following differential identities:…”

“…This is because once the infinitesimal generators in the Lie symmetry analysis are allowed to involve derivative terms then one can no longer determine the complete symmetry group. Dynamical symmetries of ODEs can be regarded as the transformations of the set of first integrals [18,19]. Deviating from the conventional approach of solving the determining equations arising from the associated invariance condition, in this paper, we develop a simple and straightforward algorithm to derive dynamical symmetries associated with the given nonlinear ODE belonging to the Riccati and Abel chains of ODEs.…”

Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

“…To overcome this demerit, during the past few years, several generalizations over the classical Lie algorithm have been proposed. Some of the algorithms that have been developed in the recent literature to derive integrals/general solution associated with the given ODE that lacks Lie point symmetries are λ-symmetries, 17 telescopical symmetry, 18 hidden and non-local symmetries, 19 adjoint symmetry method, 20 exponential vector fields, 21 generalized Lie symmetries, 22 and so on. In this paper, we intend to study nonlocal symmetries associated with the given equation.…”