Extended integrability and bi-Hamiltonian systems

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 T-q Or wind on toroidal cylinders T-m x Rq-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 T-q of arbitrary dimensions q = 1, ..., 2k - 1, 2k 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 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.