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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

Published online by Cambridge University Press:  17 February 2009

V. M. Gorringe  and
P. G. L. Leach
V. M. Gorringe
Affiliation:
Center for Nonlinear Studies and Department of Computational and Applied Mathematics, University of the Witwatersrand, P.O. WITS, 2050, South Africa.
P. G. L. Leach
Affiliation:
Center for Nonlinear Studies and Department of Computational and Applied Mathematics, University of the Witwatersrand, P.O. WITS, 2050, South Africa.
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Abstract

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In two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is 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.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 4 , April 1993 , pp. 511 - 522
Copyright
Copyright © Australian Mathematical Society 1993

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