  This paper studies hamiltonization of nonholonomic systems using geometric
tools. By making use of symmetries and suitable first integrals of the system,
we explicitly define a global 2-form for which the gauge transformed
nonholonomic bracket gives rise to a new bracket on the reduced space codifying
the nonholonomic dynamics and carrying an almost symplectic foliation
(determined by the common level sets of the first integrals). In appropriate
coordinates, this 2-form is shown to agree with the one previously introduced
locally in [34]. We use our coordinate-free viewpoint to study various
geometric features of the reduced brackets. We apply our formulas to obtain a
new geometric proof of the hamiltonization of a homogeneous ball rolling
without sliding in the interior side of a convex surface of revolution using
our formulas.
