First Integral and Stability of Motion in the Critical Cases

In this paper,we indicate that after the Liapunov function by using linear combination of mechanical first integral was suggested by Chetayev in 1946. He and his students solved stability of conservative system by means of this method. But he had trouble to solve the problems by means of cut and try. Moreover, the condition of stability is imperfect. Solution by this method is limitedfor problems of purely imaginary roots. The cases of zero roots have not been considered. Condition of stability secured is more strict.This paper suggests that the differential equation can be transformed into standard form by method of cancellation of cyclic coordinates(method of lowering degree of order), and condition of stability can be determined by energy integral. By this method not only the computation is clear and concise. But also zero roots can be considered. Therefore the problems of two cyclic coordinates can be transformed into second-order system, and we get new conclusion of the condition of stability simply. As for problems of single cyclic coordinate, in fact, Chetayev and his students did not solve the stability of the gyroscope of outer-gimbal with horizontal axis or arbitrary angle. In this paper, it shows that the method suggested here is useful for stability of these problems. The condition of conditional stability and instability were derived.