Abstract
In this paper, the integrable classical case of the Hydrogen atom subjected to three static external fields is investigated. The structuring and evolution of the real phase space are explored. The bifurcation diagram is found and the bifurcations of solutions are discussed. The periodic solutions and their associated periods for singular common-level sets of the first integrals of motion are explicitly described. Numerical investigations are performed for the integrable case by means of Poincare surfaces of section and comparing them with nearby living nonintegrable solutions, all generic bifurcations that change the structure of the phase space are illustrated; the problem can exhibit regularity-chaos transition over a range of control parameters of system.
Highlights
Most natural phenomena are generally governed by nonlinear differential equations, for these problems, we must use different mathematical approaches and computational methods to simplify these systems and to study their integrable like lie algebra, the Painlevé criterion, the Ziglin criterion, the Liouville theorem and the Poincaré sections, because physically integrable systems are rare.The study of the Hydrogen atom has been recently the focus of many works [1]-[10]
Numerical investigations are performed for the integrable case by means of Poincaré surfaces of section and comparing them with nearby living nonintegrable solutions, all generic bifurcations that change the structure of the phase space are illustrated; the problem can exhibit regularity-chaos transition over a range of control parameters of system
And α,λ,γ and β are control parameters representing the magnitude of the applied external fields: α is a constant; γ, measured in units of magnetic field = B0 2.35×105 T, controls the quadratic Zeeman effect; β, measured in units of electric field = F0 5.14 ×1011 V cm, controls the Stark effect; and the a-dimensional number λ controls the anharmonicity associated to the van der Waals interaction [10] [11] [12]
Summary
Most natural phenomena are generally governed by nonlinear differential equations, for these problems, we must use different mathematical approaches and computational methods to simplify these systems and to study their integrable like lie algebra, the Painlevé criterion, the Ziglin criterion, the Liouville theorem and the Poincaré sections, because physically integrable systems are rare. And α,λ,γ and β are control parameters representing the magnitude of the applied external fields: α is a constant; γ, measured in units of magnetic field = B0 2.35×105 T , controls the quadratic Zeeman effect; β, measured in units of electric field = F0 5.14 ×1011 V cm , controls the Stark effect; and the a-dimensional number λ controls the anharmonicity associated to the van der Waals interaction [10] [11] [12] This combined potential is very interesting because of its similarity to the fields seen by an ion confined in a Paul trap [13].
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