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The phenomenon of a topological monodromy in integrable Hamiltonian andnonholonomic systems is discussed. An efficient method for computing andvisualizing the monodromy is developed. The comparative analysis of thetopological monodromy is given for the rolling ellipsoid of revolution problemin two cases, namely, on a smooth and on a rough plane. The first of thesesystems is Hamiltonian, the second is nonholonomic. We show that, from theviewpoint of monodromy, there is no difference between the two systems, andthus disprove the conjecture by Cushman and Duistermaat stating that thetopological monodromy gives a topological obstruction for Hamiltonization ofthe rolling ellipsoid of revolution on a rough plane.

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?