We present a novel extremum seeking method for affine connection mechanical
control systems. The proposed control law involves periodic perturbation
signals with sufficiently large amplitudes and frequencies. A suitable
averaging analysis reveals that the solutions of the closed-loop system
converge locally uniformly to the solutions of an averaged system in the
large-amplitude high-frequency limit. This in turn leads to the effect that
stability properties of the averaged system carry over to the approximating
closed-loop system. Descent directions of the objective function are given by
symmetric products of vector fields in the averaged system. Under suitable
assumptions, we prove that minimum points of the objective function are
asymptotically stable for the averaged system and therefore practically
asymptotically stable for the closed-loop system. We illustrate our results by
examples and numerical simulations