Collinear Classical Helium Atom
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Collinear helium is a hypothetical system with two electrons arranged along a line on opposite sides of a nucleus of charge +2, with infinite nuclear mass. The electron positions then imply a problem with two degrees of freedom. The plots show the dynamics projected onto the plane of the position of the two electrons . For a number of initial conditions, trajectories escape to infinity after several reflections.
Contributed by: Enrique Zeleny (March 2013)
Open content licensed under CC BY-NC-SA
The Hamiltonian is
where the are the generalized momenta, and replacing
the Hamiltonian takes the form
and the equations of motion are
where prime denotes derivative with respect to time.
 P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, N. Whelan, and A. Wirzba, "Helium Atom," in Chaos: Classical and Quantum, Copenhagen: Niels Bohr Institute, 2012. www.chaosbook.org.
 K. Richter, G. Tanner, and D. Wintgen, "Classical Mechanics of Two-Electron Atoms," Physical Review A, 48(6), 1993 pp. 4182–4196. doi:10.1103/PhysRevA.48.4182.
 G. Tanner and K. Richter. "Semiclassical Theory of Helium Atom." Scholarpedia. (Jan 8, 2013) www.scholarpedia.org/article/Semiclassical_theory_of _helium _atom.