Holonomic systems on a rapidly oscillating platform

The equations of motion of holonomic systems situated on an oscillating platform in the limit as the frequency of the oscillations tends to infinity have been obtained using an averaging method. For autonomous holonomic systems located in a uniform gravitational field on a vertically oscillating platform, it is shown that all of the equilibria existing for the system in the absence of oscillations are preserved in the limit system, and all such non-degenerate equilibria of the limit system are stable for a sufficiently large amplitude of the oscillations velocity. An estimate of the amplitude of the oscillations velocity is obtained at which stabilization of unstable equilibria begins. A similar statement is valid for time-independent motions of systems with cyclic coordinates. Examples are provided.