On stability of stationary motion of a nonconservative nonholonomic rheonomic system (Q2459472)

The author studies the motion of a mechanical system in a nonstationary field of potential and positional forces, where the rheonomic form of its Lagrangian function nontransformed with respect to the equations of nonholonomic constraints results as a consequence of introducing rheonomic holonomic constraints in it. The rheonomic character of the Lagrangian function can be, for example, the consequence of influences of the given finite equations of transport motion of the mechanical system, where its relative motion is restricted by constraints which do not depend on time explicitly. In assumption that the governing differential equations satisfy the conditions of the existence of Painleve's energy integral the original mechanical system is changed by equivalent one with Lagrangian function, nontransformed relative to nonholonomic constraints non-depended explicitly on time. At the usage of properties of equivalent system, which moves in stationary (nonstationary) field of potential forces (gyroscopic forces) the authors generalize the definition of cyclic coordinates together with sufficient conditions of the first integral existence, which are linear in velocities. On this way the Ronth's theorem on stability of steady motion for conservative systems is extended on nonconservative ones.