EXTENDED INTEGRABILITY AND BI-HAMILTONIAN SYSTEMS

The current notion of integrability of Hamiltonian systems was fixed b y Liouville in a famous 1855 paper. It describes systems in a 2k-dimen sional phase space whose trajectories are dense on tori T-q Or wind on toroidal cylinders T-m x Rq-m. Within Liouville's construction the di mension 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 T-q of arbitrary dimensions q = 1, ..., 2k - 1, 2 k and an additional invariant upsilon: 2(q - k) less than or equal to upsilon less than or equal to 2[q/2] can be recovered. The main theore m classifies all k(k + 1)/2 canonical forms of Hamiltonian systems tha t are integrable in a newly defined broad sense. An integrable physica l problem having engineering origin is presented. The notion of extend ed compatibility of two Poisson structures is introduced. The correspo nding bi-Hamiltonian systems are shown to be integrable in the broad s ense.