On bifurcation and stability of stationary motions in certain problems of dynamics of a solid body: PMM vol. 38, n≗4, 1974, pp. 616–627

Abstract

Bifurcation theory for stationary motions was developed by Poincaré [1] and Chetaev [2] for Lagrangian conservative mechanical systems. This theory is based on the investigation of the (transformed) potential energy of the system V = V ( c, q 1, …, q m ), where 1, …, q m are the Lagrange coordinates and c is a parameter. For three problems in solid body dynamics we have shown below that this theory is applicable for the investigation of systems with known first integrals U ( x 1, …, x n ) = c, U 1( x 1, …, x n ) = c 1 …, U k ( x 1, …, x n ) = c k ( k + 1 < n) As in the classical case, here we can introduce the function W ( c 1,…, c k ; λ 1, …, λ k , x 1, …, x n ) = U + λ 1 ( U 1 − c 1) + … + λ k ( U k − c k ) whose role is analogous to that of potential energy in the Poincaré-Chetaev theory; here λ 1, …, λ k , x 1, …, x n formally play the role of the variables q 1, …, q m ( k + n = m).