A recent body of literature has introduced a novel approach to analysis and design of simple nonlinear control laws for mechanical systems using an energy-like function called the averaged potential. The present paper surveys the theory that has been developed over the past several years, while at the same time providing a detailed look at some new applications—principal among which are rotating heavy chains. The theory concerns averaging of Hamiltonian systems in which selected generalized velocities play the role of control inputs which are commanded to execute high-frequency oscillations. It is not yet a complete theory, and in cases where it does not characterize the stability of motions (such as in the cases we have called hovering motions), numerical methods have been used to make detailed comparisons of the averaged and nonautonomous system dynamics.