  We consider the Hamiltonian and Lagrangian formalism describing free
\k-relativistic particles with their four-momenta constrained to the
\k-deformed mass shell. We study the modifications of the formalism which
follow from the introduction of space coordinates with nonvanishing Poisson
brackets and from the redefinitions of the energy operator. The quantum
mechanics of free \k-relativistic particles and of the free \k-relativistic
oscillator is also presented. It is shown that the \k-relativistic oscillator
describes a quantum statistical ensemble with finite Hagedorn temperature. The
relation to a \k-deformed Schr\"odinger quantum mechanics in which the time
derivative is replaced by a finite difference derivative is also discussed.
