  We consider the question of existence of Hamiltonians for autonomous
non-holonomic mechanical systems in this paper. The approach is elementary in
the sense that the existence of a Hamiltonian for a given non-holonomic system
is considered to be equivalent to the existence of a non-degenerate Lagrangian
for the system in question. The possible existence of such a Lagrangian is
related to the inverse problem of constructing a Lagrangian from the
appropriate equations of motion. A simple example in three dimensions with one
non-holonomic constraint is analysed in detail, and it is shown that in this
case there is no Lagrangian reproducing the equations of motion in three
dimensions. Thus the system does not admit a variational formulation in three
dimensions. However, the system in question is equivalent to a two-dimensional
system which admits a variational formulation. Two distinct Lagrangians and
their corresponding Hamiltonians are constructed explicitly for this
two-dimensional system
