  In the framework of Lagrangian formulation, some q-deformed physical systems
are considered. The q-deformed Legendre transformation is obtained for the free
motion of a non-relativistic particle on a quantum line. This is subsequently
exploited to obtain the Lagrangians for the q-deformed harmonic oscillator and
q-deformed relativistic free particle. The Euler-Lagrange equations of motion
are derived in a consistent manner with the corresponding Hamilton's equations.
The Lagrangian for the q-deformed relativistic particle is endowed with the
q-deformed gauge symmetry and reparametrization invariance which are shown to
be equivalent only for $ q = \pm 1$.
