  Dynamical systems associated with a q-deformed two dimensional phase space
are studied as effective dynamical systems described by ordinary variables. In
quantum theory, the momentum operator in such a deformed phase space becomes a
difference operator instead of the differential operator. Then, using the path
integral representation for such a dynamical system, we derive an effective
short-time action, which contains interaction terms even for a free particle
with q-deformed phase space. Analysis is also made on the eigenvalue problem
for a particle with q-deformed phase space confined in a compact space. Under
some boundary conditions of the compact space, there arises fairly different
structures from $q=1$ case in the energy spectrum of the particle and in the
corresponding eigenspace .
