MONODROMY IN THE HYDROGEN ATOM IN CROSSED FIELDS
R. H. Cushman
Mathematics Institute, University of Utrecht,
Budapestlaan 6, 3508 TA Utrecht, The Netherlands
D. A. Sadovski\'{\i}
Universit\'e du Littoral, Citadelle, BP 5526, 59379 Dunkerque Cedex, France
Received 8 June 1999; revised 1 February 2000; accepted 9 February 2000
Communicated by J.D. Meiss
Available online 25 May 2000.
We show that the hydrogen atom in orthogonal electric and magnetic
fields has a special property of certain integrable classical
Hamiltonian systems known as monodromy. The strength of the fields is
assumed to be small enough to validate the use of a truncated normal
form Hsnf which is obtained from a two step normalization of the
original system. We consider the level sets of Hsnf on the second
reduced phase space. For an open set of field parameters we show that
there is a special dynamically invariant set which is a ``doubly pinched
2-torus''. This implies that the integrable Hamiltonian Hsnf has
monodromy. Manifestation of monodromy in quantum mechanics is also
discussed.
PACS: 03.20.+i Classical mechanics of discrete systems;
32.60+i Zeeman and Stark effects
Keywords: Singular reduction; Monodromy; Energy-momentum map
Physica D: Nonlinear Phenomena
Volume 142, Issues 1-2 , 1 August 2000, Pages 166-196