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Complete sets of invariants for dynamical systems that admit a separation of variables

J. Math. Phys. 43, 3592 (2002); doi:10.1063/1.1484540

Issue Date: July 2002

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E. G. Kalnins and J. M. Kress
Department of Mathematics, University of Waikato, Hamilton, New Zealand

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455

G. S. Pogosyan
Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48–3, 62251 Cuernavaca, Morelos, México
Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2,...,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = deltaij. The 2n–1 functions Q2,...,Qn,P1,...,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion. ©2002 American Institute of Physics.
History: Received 7 March 2002; accepted 20 March 2002
Permalink: http://link.aip.org/link/?JMAPAQ/43/3592/1
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KEYWORDS and PACS

Keywords
PACS
  • 45.20.Jj
    Classical mechanics of discrete systems Formalisms in classical mechanics Lagrangian and Hamiltonian mechanics
  • 45.05.+x
    Classical mechanics of discrete systems General theory of classical mechanics of discrete systems
  • YEAR: 2002

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0022-2488 (print)   1089-7658 (online)
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