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one-dimensional conservative system with position depending mass, one deduces
consistently a constant of motion, a Lagrangian, and a Hamiltonian for the
nonrelativistic 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 cannot 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.