The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation. This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces. The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system. The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method. The results are compared with those for the classical anharmonic oscillator. This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator. It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method. It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.

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The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation. This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space. These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process. The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator. These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system. The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method. The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h. This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator. The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator. These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator. This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.

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