  We examine the problem of defining Lagrangian and Hamiltonian mechanics for a
particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we
first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting
particle coordinates and velocities. Using techniques similar to those of Wess
and Zumino, we construct two different differential calculi on $TQ_{q,p}$.
These two differential calculi can in principle give rise to two different
particle dynamics, starting from a single Lagrangian. For Hamiltonian
mechanics, we define a phase space $T^*Q_{q,p}$ spanned by noncommuting
particle coordinates and momenta. The commutation relations for the momenta can
be determined only after knowing their functional dependence on coordinates and
velocities.
  Thus these commutation relations, as well as the differential calculus on
$T^*Q_{q,p}$, depend on the initial choice of Lagrangian. We obtain the
deformed Hamilton's equations of motion and the deformed Poisson brackets, and
their definitions also depend on our initial choice of Lagrangian. We
illustrate these ideas for two sample Lagrangians. The first system we examine
corresponds to that of a nonrelativistic particle in a scalar potential. The
other Lagrangian we consider is first order in time derivatives
