We present a geometric construction of almost Poisson brackets for nonholonomic mechanical systems whose configuration space is a Lie group G. We study the so-called LL and LR systems where the kinetic energy defines a left invariant metric on G and the constraints are invariant with respect to left (respectively right) translation on G.For LL systems, the equations on the momentum phase space, T*G, can be left translated onto g*, the dual space of the Lie algebra g. We show that the reduced equations on g* can be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the standard Lie-Poisson bracket onto the constraint space.For LR systems, we use ideas of semidirect product reduction to transfer the equations on T*G into the dual Lie algebra, s*, of a semidirect product. This provides a natural Lie algebraic setting for the equations of motion commonly found in the literature. We show that these equations can also be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the Lie-Poisson structure on s* onto a constraint submanifold.In both cases the constraint functions are Casimirs of the bracket and are satisfied automatically. Our construction is a natural generalization of the classical ideas of Lie-Poisson and semidirect product reduction to the nonholonomic case. It also sets a convenient stage for the study of Hamiltonization of certain nonholonomic systems.Our examples include the Suslov and the Veselova problems of constrained motion of a rigid body, and the Chaplygin sleigh.In addition we study the almost Poisson reduction of the Chaplygin sphere. We show that the bracket given byBorisov and Mamaev is obtained by reducing a nonstandard almost Poisson bracket that is obtained by projecting a non-canonical bivector onto the constraint submanifold using the Lagrange-D'Alembert principle.The examples that we treat show that it is possible to cast the reduced equations of motion of certain nonholonomic systems in Hamiltonian form (in the Poisson formulation) either by multiplication by a conformal factor, by the use of nonstandard brackets or simply by reduction methods.

In this Thesis we study two types of mechanical nonholonomic systems, namely systems with linear constraints and lagrangian with a linear term in the velocities, and nonholonomic systems with affine constraints and lagrangian without a linear term in the velocities. For the former type of systems we construct an almost-Poisson bracket using elements related to a riemannian metric induced by the kinetic energy, and we show that under certain conditions gauge momenta exist. For the latter type of systems, we focus on the ones possessing a \emph{Noether symmetry}. To everyone of these systems we associate an equivalent system of the former type, and we exhibit the procedure to relate them and their gauge momentum. As a test case for the theory, we analyze the system of a heavy ball rolling without slipping on a rotating surface of revolution: we elucidate that also in this framework the so-called Routh integrals are related to symmetries, we give conditions for boundedness of the motions. In the particular case the surface of revolution is an inverted cone we characterize the qualitative behavior of the motions.