On stability of nonconservative systems
The author considers the system \(M\ddot y+ K_1y+ P_2y=\widetilde F(y)\), where \(M= M^T> 0\), \(K_1= K^T_1\), \(P_1= -P^T_1\) are matrices of dimension \(n\times n\), \(\widetilde F(y)\) is the matrix-column of terms which are nonlinear with respect to \(y\) and arise from \(F_1= \text{grad }\widetilde U(y)\) and \(F_2\), and moreover, the conditions \(y^T F_2= 0\), \(\widetilde F(0)= 0\) hold. The author presents this system in the form \[ \ddot x+ Kx+ Px= Q(x)\tag{1} \] with \(K= K^T= 1/2(M^{-1}(K_1+ P_1)+ (K_1- P_1)M^{-1})\), \(P= -P^T= 1/2(M^{-1}(K_1+ P_1)- (K- P)M^{-1})\), \(Q(x)= \widetilde Q_1+\widetilde Q_2(x)\), \(\widetilde Q_1= \text{grad }U(x)\), \(x^T\widetilde Q_2= 0\), \(Q(0)= 0\). A few theorems on instability of the equilibrium of (1) in the domain of system parameters are proved. (For example, Theorem 1: If \(K<0\), the equilibrium of the nonconservative system (1) is unstable independently on nonlinear forces, is positional so depending on \(\dot x\). Theorem 2: If \(K= 0\), \(x^T\text{grad }U(x)< 0\), the equilibrium of (1) is unstable under any nonconservative positional forces.) For proving theorems, the author uses the differential consequence to system (1): \({d\over dt} f(\dot x,x)= F(\dot x,x)\).
scientific article; zbMATH DE number 1309948