Integrability in the broad sense
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 Tq or wind on toroidal cylinders Tm×Rq-m. Within Liouville's construction the dimension q cannot exceed k and is the main invariant of the system. In the talk, I will present a generalization of Liouville integrability so that trajectories can be dense on tori Tq 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 in the broad sense physical problem having engineering origin will be presented.