Communications in Mathematical Physics

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

Extended Integrability and Bi-Hamiltonian Systems

  • Oleg I. Bogoyavlenskij


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.


Phase Space Hamiltonian System Broad Sense Canonical Form Physical Problem 
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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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