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Dynamical systems associated with a q-deformed two dimensional phase spaceare studied as effective dynamical systems described by ordinary variables. Inquantum theory, the momentum operator in such a deformed phase space becomes adifference operator instead of the differential operator. Then, using the pathintegral representation for such a dynamical system, we derive an effectiveshort-time action, which contains interaction terms even for a free particlewith q-deformed phase space. Analysis is also made on the eigenvalue problemfor a particle with q-deformed phase space confined in a compact space. Undersome boundary conditions of the compact space, there arises fairly differentstructures from $q=1$ case in the energy spectrum of the particle and in thecorresponding eigenspace .

A Dynamical System with Q-deformed Phase Space Represented in Ordinary Variable Spaces