Hamiltonization of nonholonomic systems
  We consider some issues of the representation in the Hamiltonian form of two problems of nonholonomic mechanics, namely, the Chaplygin's ball problem and the Veselova problem. We show that these systems can be written as generalized Chaplygin systems and can be integrated by the method of reducing multiplier. We also indicate the algebraic form of the Poisson brackets of these systems (after the time substitution). Generalizations of the problems are considered and new realizations of nonholonomic constraints are presented. Some nonholonomic systems with an invariant measure and a sufficient number of first integrals are indicated, for which the question of the representation in the Hamiltonian form is still open, even after the time substitution. 
A.V. Borisov, I.S. Mamaev