The instability of the equilibrium of a two-degree-of-freedom gyro
Consider an analytical mechanical system with the Lagrangian function \[ (1)\quad L=(\dot q_ 1^ 2;\dot q^ 2_ 2)+\omega (q_ 1\dot q_ 2- q_ 2\dot q_ 1)+V(q_ 1,q_ 2). \] Lagrangian equations with the Lagrangian (1) can be, for example, equations of motion of a material point on a plane, acted upon by certain potential forces and a magnetic force, with the vector of magnetic induction perpendicular to the plane of motion. Let \(q_ 1=q_ 2=0\) be the equilibrium position of the system, and suppose that the reduced force function V can be expanded as a MacLaurin series \(V(q_ 1,q_ 2)=\sum^{\infty}_{k=0}V_{m+k}(q_ 1,q_ 2),\) \(m\geq 2\), where \(V_{m+k}\) are homogeneous polynomials of degree \(m+k\). If \(\omega =0\) and the form \(V_ m\) can take positive values, the equilibrium position \(q_ 1=q_ 2=0\) is unstable. If, however, \(\omega\neq 0\), the gyroscopic stabilization is possible, and it is of interest to investigate sufficient conditions of instabilities, some of these conditions are stated and proved in this paper.
scientific article; zbMATH DE number 4081004