The averaging method, a pendulum with a vibrating suspension: N. N. Bogolyubov, A. Stephenson, P. L. Kapitza and others

The main moments of the historical development of one of the basic methods of nonlinear systems investigating (the averaging method) are traced. This method is understood as a transition from the so-called exact equation $dx/dt = \varepsilon X ( t, x ) $ ($\varepsilon$ is small parameter), to the averaging equation $d\xi/ dt = \varepsilon X_0(\xi) + \varepsilon^2 P_2(\xi) + ... + \varepsilon^m P_m(\xi)$ by corresponding variable substitution. Bogolyubov-Krylov’s approach to the problem of justifying the averaging method, based on the invariant measure theorem, is analyzed. The paper presents the evolution of views on a physical pendulum with a vibrating suspension, beginning with the description of its simple motions (A. Stephenson, G. Jeffreys, N.N. Bogolyubov, P.L. Kapitza, V.N. Chelomey, etc.) and ending with complex movements. In the latter case, various characteristic features of the complex behavior of nonlinear systems is appeared - bifurcations, chaotic regimes, etc., (J. Blackburn, M. Bartuccelli, and others). A number of analogs of a pendulum with a vibrating suspension point outside of classical mechanics are described (A.V. Gaponov, M.A. Miller - localization of a particle in an electric field; S.M. Osovets - stabilization of hot plasma; V. Paul, N. Ramsey, H. Dehmelt - confinement of particles in an alternating electromagnetic field). An important part of the work is historical information about N.M. Krylov, N.N. Bogolyubov, P.L. Kapitza, which makes possible to more clearly show the motivation of the studies, their conditionality.