Dynamics of atomic hydrogen in a generalized van der Waals potential

Generalized van der Waals Hamiltonian: periodic orbits and C1 nonintegrability

The aim of this paper is to study the periodic orbits of the generalized van der Waals Hamiltonian system. The tool for studying such periodic orbits is the averaging theory. Moreover, for this Hamiltonian system we provide information on its C(1) nonintegrability, i.e., on the existence of a second first integral of class C(1).

On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

Coupled second-order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the method of deriving a general solution for two coupled second-order nonlinear ODEs through the extended Prelle–Singer procedure. We describe a procedure to obtain integrating factors and the required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above-mentioned method, we introduce a new method of transforming coupled second-order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.

Hydrogen atom in the presence of uniform magnetic and quadrupolar electric fields: integrability, bifurcations, and chaotic behavior

We investigate the classical dynamics of a hydrogen atom in the presence of uniform magnetic and quadrupolar electric fields. After some reductions, the system is described by a two degree of freedom Hamiltonian depending on two parameters. On the one hand, it depends on the z component of the canonical angular momentum P(phi), which is an integral because the system is axially symmetric; and on the other it also depends on a parameter representing the relative field strengths. We note that this Hamiltonian is closely related to the one describing the generalized van der Waals interaction. We report three cases of integrability. The structure and evolution of the phase space are explored intensively by means of Poincaré surfaces of section when the parameters vary. In this sense, we find several bifurcations that strongly change the phase space structure. The chaotic behavior of the system is studied and three order-chaos transitions are found when the system passes through the integrable cases. Finally, the ionization mechanics is studied.