Geometry of Lax pairs: Particle motion and Killing-Yano tensors

Marco Cariglia, Valeri P. Frolov, Pavel Krtouš, and David Kubizňák
Phys. Rev. D 87, 024002 – Published 2 January 2013

Abstract

A geometric formulation of the Lax pair equation on a curved manifold is studied using the phase-space formalism. The corresponding (covariantly conserved) Lax tensor is defined and the method of generation of constants of motion from it is discussed. It is shown that when the Hamilton equations of motion are used, the conservation of the Lax tensor translates directly to the well-known Lax pair equation, with one matrix identified with components of the Lax tensor and the other matrix constructed from the (metric) connection. A generalization to Clifford objects is also discussed. Nontrivial examples of Lax tensors for geodesic and charged particle motion are found in spacetimes admitting a hidden symmetry of Killing-Yano tensors.

  • Figure
  • Received 22 October 2012

DOI:https://doi.org/10.1103/PhysRevD.87.024002

© 2013 American Physical Society

Authors & Affiliations

Marco Cariglia*

  • Departamento de Física, Universidade Federal de Ouro Preto, ICEB, Campus Morro do Cruzeiro, Ouro Preto 35400-000, Minas Gerais, Brasil

Valeri P. Frolov

  • Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2G7

Pavel Krtouš

  • Faculty of Mathematics and Physics, Institute of Theoretical Physics, Charles University in Prague, V Holešovičkách 2, 180 00 Prague, Czech Republic

David Kubizňák§

  • Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

  • *marco@iceb.ufop.br
  • vfrolov@ualberta.ca
  • Pavel.Krtous@utf.mff.cuni.cz
  • §dkubiznak@perimeterinstitute.ca

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Vol. 87, Iss. 2 — 15 January 2013

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