Extended Integrability and Bi-Hamiltonian Systems
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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(q−k) ≤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.
KeywordsPhase Space Hamiltonian System Broad Sense Canonical Form Physical Problem
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