The stabilization of the equilibrium of conservative systems using gyroscopic forces (Q2761256)

The author considers a conservative holonomic system with \(n\) degrees of freedom \(\frac{d}{dt}\frac{\partial L}{\partial\dot q}- \frac{\partial L}{\partial q} = 0\), where \(L(q,\dot q) = L_2(q,\dot q) + L_1(q,\dot q) + L_0(q) = \frac 12 \dot q^TA(q)\dot q+f(q)^T\dot q+L_0(q)\). It is assumed that \(L(q,\dot q)\in C^2(D_q\times \mathbb{R}^n)\); the quadratic form \(L_2(0,\dot q)\) is positive definite, the point \(q = \dot q = 0\) corresponds to the equilibrium state of the system, and \(f(0) = L_0(0) = 0\). Under some additional assumptions the author derives instability conditions for the equilibrium of the system under gyroscopic forces.