Lagrangian Formalism for a Classical Particle Moving in Riemannian Space with Dissipative Forces

Equations of motion are generated for a classical particle in a Riemannian space with arbitrary metric, moving in a velocity-independent potential, and acted upon by a linear frictional force. Only the Lagrangian formalism is used, but the Lagrangian function is not of standard form. As an example, a satellite moving in a linearly-viscous atmosphere is considered. This approach yields immediately a constant of the motion which is a generalization of the usual angular momentum. A particle moving on the viscous surface of a sphere is also discussed. Finally, the equation of motion for a particle moving in one dimension subject to a frictional force quadratic in the velocity is generated geometrically by choosing an appropriate nonanalytic metric.