  On the basis of the non-commutative q-calculus, we investigate a
q-deformation of the classical Poisson bracket in order to formulate a
generalized q-deformed dynamics in the classical regime. The obtained
q-deformed Poisson bracket appears invariant under the action of the
q-symplectic group of transformations. In this framework we introduce the
q-deformed Hamilton's equations and we derive the evolution equation for some
simple q-deformed mechanical systems governed by a scalar potential dependent
only on the coordinate variable. It appears that the q-deformed Hamiltonian,
which is the generator of the equation of motion, is generally not conserved in
time but, in correspondence, a new constant of motion is generated. Finally, by
following the standard canonical quantization rule, we compare the well known
q-deformed Heisenberg algebra with the algebra generated by the q-deformed
Poisson bracket.
