  A nonholonomic system consists of a configuration space Q, a Lagrangian L,
and an nonintegrable constraint distribution H, with dynamics governed by
Lagrange-d'Alembert's principle. We present two studies both using adapted
moving frames. In the first study we apply Cartan's method of equivalence to
investigate the geometry underlying a nonholonomic system. As an example we
compute the differential invariants for a nonholonomic system on a
four-dimensional configuration manifold endowed with a rank two (Engel)
distribution. In the second part we study G-Chaplygin systems. These are
systems where the constraint distribution is given by a connection on a
principal fiber bundle with total space Q and base space S=Q/G, and with a
G-equivariant Lagrangian. These systems compress to an almost Hamiltonian
system on $T^{*}S$. Under an $s \in S$ dependent time reparameterization a
number of compressed systems become Hamiltonian. A necessary condition for
Hamiltonization is the existence of an invariant measure on the original
system. Assuming an invariant measure we describe the obstruction to
Hamiltonization. Chaplygin's "rubber" sphere, a ball with unequal inertia
coefficients rolling without slipping or spinning (about the vertical axis) on
a plane is Hamiltonizable when compressed to $T^{*}SO(3)$. Finally we discuss
reduction of internal symmetries. Chaplygin's "marble" (where spinning is
allowed) is not Hamiltonizable when compressed to $T^{*}SO(3)$; we conjecture
that it is also not Hamiltonizable when reduced to $T^{*}S^{2}$.
