Potential motion in a geometric setting: presenting differential geometry methods in a classical mechanics course

Abstract

The standard classical mechanics textbooks used at graduate level mention geometrization of the potential motion kinematics. We show that the complete problem can also be geometrized, presenting the system of equations of geometric origin equivalent to the equations of motion of the potential system. The subject seems to be an excellent opportunity for introducing differential geometry concepts already in the classical mechanics course. After presenting the necessary differential geometry notions, the classical mechanical potential system is described in geometric terms. To the system one associates a Riemann space with an appropriately chosen metric and affine connection, both specified by the potential. In this picture, both dynamics and kinematics acquire invariant geometric meaning.