Periodic flows, rank-two Poisson structures, and nonholonomic mechanics

It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a rank-two Poisson structure. We link the existence of these structures to a dynamical property of the (reduced) system: its periodicity, with positive period depending continuously on the initial data. Moreover, we show that there are in fact infinitely many such Poisson structures and we classify them. We illustrate the situation on the sample case of a heavy ball rolling on a surface of revolution.