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