On the stability of stationary motions of systems with quasi-ignorable coordinates and of mechanical equilibrium under the action of a magnetic field: PMM vol. 37, n≗3, 1973, pp. 400–406

Abstract

It has been shown that the stability of the steady-state motions of systems with quasi-ignorable coordinates can be judged from the stability of the equilibrium of a position subsystem with constant (invariable) quasi-ignorable generalized velocities. This allows us to disregard the degrees of freedom corresponding to the quasi-ignorable coordinates and to use Poincare's results on the change of stability at the bifurcations of equilibria by taking the quasi-ignorable velocities as parameters. Examples of systems of the class being considered are electromechanical systems not containing capacitances. The results mentioned above are valid for them also for a nonlinear connection between B and Hin magnetics. In the case of ignorable coordinates the judgement on the stability of a stationary motion from the stability of the equilibrium of a position subsystem is possible with the aid of Routh's theorem generalized and supplemented by Liapunov [1]. However, this case differs essentially from the one being considered in that it is the ignorable momenta and not the velocities that are taken as constants.