Infinitesimally Stable and Unstable Singularities of 2-Degrees of Freedom Completely Integrable Systems
2007, vol. 12, no. 6, pp. 717-731
In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.
Fasso F., Giacobbe A., Sansonetto N.
Periodic flows, rank-two Poisson structures, and nonholonomic mechanics
2005, vol. 10, no. 3, pp. 267-284
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.