Dissipative and passive mechanical systems are studied from a geometric point of view. Since the natural geometric background is a Riemannian manifold, we begin by generalizing La Salle theorems about the stability of equilibrium points of dynamical systems to a complete Riemannian manifold. The stability of dissipative mechanical systems is studied using the particular geometric properties of the tangent bundle, and passivity based controls are designed to stabilize equilibrium points. The case of partially dissipative systems is formulated and used with a dynamical extension to design controls for bringing the system to a desired point of the phase space.