Lagrangian formulation of some q-deformed systems

In the framework of the Lagrangian formalism, 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 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 equations. The Lagrangian for the q-deformed relativistic particle is endowed with q-deformed gauge symmetry and reparametrization invariance which are shown to be equivalent only for q = ±1