Collinear Classical Helium Atom

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.


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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.
[1] 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.
[2] 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.
[3] G. Tanner and K. Richter. "Semiclassical Theory of Helium Atom." Scholarpedia. (Jan 8, 2013) _helium _atom.
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