q-Equivalent Particle Hamiltonians. I. The Classical One-Dimensional Case

The classes of equivalent Lagrangians in one-dimensional particle dynamics are found. These classes contain not only Lagrangians yielding the same equations of motion (Lagrangians differing by a total time derivative), but also those implying each other's equations of motion. The corresponding classes of Hamiltonians, all of which give the same orbits in configuration space, but in general different orbits in phase space, are also found. Some specific examples are presented.