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About One-Dimensional Conservative Systems with Position Depending Mass

For a one-dimensional conservative systems with position depending mass, one deduces consistently a constant of motion, a Lagrangian, and a Hamiltonian for the non relativistic case. With these functions, one shows the trajectories on the spaces $(x,v)$ and ($x,p)$ for a linear position depending mass. For the relativistic case, the Lagrangian and Hamiltonian can not be given explicitly in general. However, we study the particular system with constant force and mass linear dependence on the position where the Lagrangian can be found explicitly, but the Hamiltonian remains implicit in the constant of motion.