On steady-state motions of a mechanical system with the partial Steklov’s integrals

The Routh-Lyapunov’s method has been applied to find steady-state motions of a conservative autonomous mechanical system, for which the existence of an additional partial V. A. Steklov’s integral is possible. Depending on the number of integrals involved in the bundle of first integrals according to the Routh-Lyapunov method, several properties have been determined. In particular, it is shown that in case of a consequent lowering of the number of integrals in the Lagrange function, analysis of stationary solutions is conducted with fewer computational operations. Under all the methods of constructing the bundle of integrals, the stationary motions are the same. The last property formulates the theorem, in accordance with which the Lagrange function (with the minimum number of participating integrals) is obtained, when the Steklov integral is not included in it, but under the conditions of fulfillment of the integrals arising in the case of Steklov’s conditions: q = const, r = 0. In case of such a construction of the bundle of integrals, the stationary solutions exactly coincide with the steady-state motions.