Superintegrable systems, multi-Hamiltonian structures and Nambu mechanics in an arbitrary dimension

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TEĞMEN A., Vercin A.

INTERNATIONAL JOURNAL OF MODERN PHYSICS A, vol.19, no.3, pp.393-409, 2004 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 3
  • Publication Date: 2004
  • Doi Number: 10.1142/s0217751x04017112
  • Journal Name: INTERNATIONAL JOURNAL OF MODERN PHYSICS A
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.393-409
  • Keywords: superintegrable systems, multi-Hamiltonian structures, Nambu mechanics, GEOMETRIC QUANTUM-MECHANICS, POISSON, DYNAMICS, BRACKETS, ALGEBRAS, FIELDS
  • Ankara University Affiliated: Yes

Abstract

A general algebraic condition for the functional independence of 2n - 1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero-Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodinsky-Winternitz potentials are considered as illustrative applications and their symmetry algebras as well as their Nambu formulations and alternative Poisson structures are presented.