Geometric and dynamic phase-space structure of a class of nonholonomic integrable systems with symmetries

This work studies a certain class of nonholonomic mechanical systems: a sphere which rolls without sliding inside a rotating convex surface. We study the dynamics of the system, performing some reductions by symmetry, and we consider the reduced system which has 3 first integrals. We prove that these 3 first integrals are functionally independent and we use them to show that, for a certain choice of the surface profile, the reduced dynamics is periodic and the ureduced dynamics is quasi-periodic. Next we consider the case in which the surface is a known profile, a paraboloid of revolution, and we explicitly build the 3 first integrals in order to use them to have a better understanding of the reduced dynamics. We restrict the reduced dynamics to the common level sets of 2 of the first integrals: we show that the restricted systems are 2-parameters Lagrangian systems with 1 degree of freedom. Next we realize a numerical analysis to study the equilibria of this family of Lagrangian systems and we show the appearance of stable and unstable equilbria with a bifurcation mechanism, as we change the values of the parameters.