THE ASYMPTOTIC MOTIONS OF SYSTEMS WITH DISSIPATION

The problem of the existence and analytical properties of asymptotic s olutions of the equations of dynamics which approach a position of equ ilibrium as t --> infinity is considered. This problem was solved by L yapunov [1] in the case when the equilibrium is a non-degenerate criti cal point of potential energy. In this paper we consider the situation when the absence of a minimum of the potential energy cannot be deter mined from the quadratic form of the expansion of the potential energy in a Taylor series. It is shown that the asymptotic solutions can be obtained in the form of a series in inverse powers of time, which cont ains logarithms. If these series diverge, they are asymptotic expansio ns for the solutions considered. The problem of the effect of gyroscop ic forces on the existence of asymptotic motions in systems with a deg enerate potential energy is considered. An analogue of Kelvin's theore m is obtained on the impossibility of stabilizing the equilibrium by g yroscopic and dissipative forces.