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Abstract

We treat the classical dynamics of the hydrogen atom in perpendicular electric and magnetic fields as a celestial mechanics problem. By expressing the Hamiltonian in appropriate action-angle variables, we separate the different time scales of the motion. The method of averaging then allows us to reduce the system to two degrees of freedom, and to classify the most important periodic orbits.

Complete classification of qualitatively different perturbations of the hydrogen atom in weak near-orthogonal electric and magnetic fields

“…The same approach can be generalized to three-dimensional lattices. In that way, we can confirm the results of the classical analysis in section 2.3 for M in (9) and (10). Note that up to the natural increase in the number of nodes near the boundaries, quantum lattices for all systems extend trivially in the third dimension n. This is in agreement with the classical result k N = 0 in section 2.3.…”

“…Monodromy matrix M(k) (see(9) for T 3 ⊂ R 6 in systems of different strata(cf table 3). For systems of type A 1,1 we distinguish additionally monodromy matrices corresponding to the circuits [ + ], [ − ] and [ − ] + [ −…”

Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

“…Unfortunately, this aspect remained underdeveloped by Cushman & Sadovskií(1999, Efstathiou et al (2004) and has not been appreciated duly. Without any appropriate framework and correct methodology, physicists were confined to very incomplete studies (Flöthmann et al 1994;von Milczewski & Uzer 1997;Main et al 1998;Berglund & Uzer 2001;Gekle et al 2006). So one of our main goals here was to spell out the general approach to the classification of systems with Hamiltonian (1.1), based on the two-step normalization, the equivalence relation in definition 2.2, the appropriate choice of parameters, and the zone structure of the parameter space.…”

The Stark problem in the Weierstrassian formalism

“…From a purely mathematical perspective, the importance of the Stark problem lies mainly in fact that it belongs to the very restrictive class of Liouville-integrable dynamical systems of classical mechanics (Arnold 1989). Action-angle variables for the Stark problem can be introduced in a perturbative fashion, as explained in Born (1927) and Berglund & Uzer (2001).…”

“…Different types of solutions to the Stark problem are available in the literature. If the constant acceleration field is much smaller than the Keplerian attraction along the orbit of the test particle, the problem can be treated in a perturbative fashion, and the (approximate) solution is expressed as the variation in time of the Keplerian (or Delaunay) orbital elements of the osculating orbit (Vinti 1966;Berglund & Uzer 2001;Namouni & Guzzo 2007;Belyaev & Rafikov 2010;Pástor 2012). A different approach is based on regularisation procedures such as the Levi-Civita and Kustaanheimo-Stiefel transformations (Kustaanheimo & Stiefel 1965;Saha 2009), which yield exact solutions in a set of variables related to the cartesian ones through a rather complex nonlinear transformation (Kirchgraber 1971;Rufer 1976;Poleshchikov 2004).…”