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Communications in Mathematical Physics

, Volume 196, Issue 1, pp 19–51 | Cite as

Extended Integrability and Bi-Hamiltonian Systems

  • Oleg I. Bogoyavlenskij

Abstract:

The current notion of integrability of Hamiltonian systems was fixed by Liouville in a famous 1855 paper. It describes systems in a 2k-dimensional phase space whose trajectories are dense on tori ? q or wind on toroidal cylinders ? m ×ℝ q-m . Within Liouville's construction the dimension q cannot exceed k and is the main invariant of the system. In this paper we generalize Liouville integrability so that trajectories can be dense on tori ? k of arbitrary dimensions q= 1, …, 2k−1, 2k and an additional invariant v: 2(qk) ≤v≤ 2[q/2] can be recovered. The main theorem classifies all k(k+1)/2 canonical forms of Hamiltonian systems that are integrable in a newly defined broad sense. An integrable physical problem having engineering origin is presented. The notion of extended compatibility of two Poisson structures is introduced. The corresponding bi-Hamiltonian systems are shown to be integrable in the broad sense.

Keywords

Phase Space Hamiltonian System Broad Sense Canonical Form Physical Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Oleg I. Bogoyavlenskij
    • 1
  1. 1.Department of Mathematics and Statistics, Queen's University, Kingston, Canada, K7L 3N6CA

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