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

      Kalnins, Ernie G.; Kress, Jonathan M.; Miller, W., Jr.; Pogosyan, G.S.
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      Kalnins complete sets.pdf
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      DOI
       10.1063/1.1484540
      Link
       link.aip.org
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      Kalnins, E.G., Kress, J.M., Miller, W., Jr. & Pogosyan, G.S. (2002). Complete sets of invariants for dynamical systems that admit a separation of variables. Journal of Mathematical Physics, 43, 3592.
      Permanent Research Commons link: https://hdl.handle.net/10289/1189
      Abstract
      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} = ij. 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.
      Date
      2002-07
      Type
      Journal Article
      Publisher
      American Institute of Physics
      Rights
      Copyright 2002 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in the Journal of Mathematical Physics and may be found at http://jmp.aip.org/jmp/top.jsp
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