Non-Integrability and Structure of the Resonance Zones in a Class of Galactic Potentials

Abstract

The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more general method than the pendulum approximation. This method applies to the case of a non-degenerate integrable part in the Hamiltonian. This problem may be overcome in a class of galactic-type polynomial potentials, in the case where the higher-order term is by itself integrable. An illustrative example is worked out.

Periodic orbits and non-integrability of Hénon–Heiles systems

“…Another interesting criterion on non-integrability also related to the Poincaré result was used by Meletlidou and Ichtiarouglou [21][22][23]. They considered perturbed Hamiltonian systems of the form H = H 0 + εH 1 , where H 0 is a non-degenerate integrable Hamiltonian, and showed that some properties of the average value of the perturbing function H 1 , evaluated along the non-isolated periodic orbits of H 0 , are strongly connected with the non-integrability of the perturbed system.…”

Periodic orbits of the generalized Friedmann-Robertson-Walker potential in galactic dynamics in a rotating reference frame

“…1 In the last years, the most of work on galactic dynamic has been restricted to the models of elliptical galaxies (for more details see, e.g., Ref. [2][3][4][5][6][7][8]. The majority of galactic potential has symmetry about the two axes and thus, it can be written in the form V (x 2 , y 2 ).…”

Dynamics of a galactic Hamiltonian system

“…Another interesting criterion on non-integrability also related with the Poincaré's result was used by Meletlidou and Ichtiarouglou [11,12,13]. They consider perturbed Hamiltonian systems of the form H = H 0 + εH 1 , where H 0 is a non-degenerate integrable Hamiltonian, and they show that some properties of the averaged value of the perturbing function H 1 , evaluated along the non-isolated periodic orbits of H 0 , are strongly connected with the non-integrability of the perturbed system.…”