Hamiltonian symmetry classification, integrals, and exact solutions of a generalized Ermakov system

摘要

We investigate the Hamiltonian symmetry classification, first integrals, and exact solutions of the three-dimensional generalized Ermakov system expressed in Spherical polar form. We utilize the Hamiltonian version of Noether's theorem to construct Hamiltonian symmetries and first integrals. The three-dimensional generalized Ermakov system involves an arbitrary function F(theta, phi). First, the Hamiltonian symmetry classification is performed for the three-dimensional generalized Ermakov system. For arbitrary form of the function F(theta, phi), four first integrals are established, and only three of these are functionally independent. The Hamiltonian integral theorem yields seven different functional forms of F(theta, phi) for which the generalized Ermakov system has additional Hamiltonian symmetries and related first integrals. These functional forms are new and not provided before in the literature. Finally, the exact solutions of the generalized Ermakov system, for arbitrary F(theta, phi) as well as for different forms of F(theta, phi) are provided with the aid of these derived first integrals.