J Austral Math Soc Ser B 34 pp511--522, 1993.

The first integrals and their Lie algebra of the most general autonomous Hamiltonian of the form H = T + V possessing a Laplace-Runge-Lenz vector

V. M. Gorringe and P. G. L. Leach

Abstract

In two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is H = ½ p·p - mr^-1 - ar^-½cos((q - b)/2). The Poisson bracket of the two components of this vector leads to a third first-integral, cubic in the momenta. The Lie algebra of the three integrals under the operation of the Poisson bracket closes, and is shown to be so(3) for negative energy and so(2, 1) for positive energy. In the case of zero energy, the algebra is W(3,1). The result does not have a three-dimensional analogue, apart from the usual Kepler problem.

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Authors

V. M. Gorringe

P. G. L. Leach

Centre for Nonlinear Studies and Department of Computational and Applied Mathematics, University of the Witwatersrand, P.O. WITS, 2050 South Africa.