On stability of motion of the Maxwell pendulum

We investigate the stability of motion of the Maxwell pendulum in a uniform gravity field [1, 2]. The threads on which the axis and the disk of the pendulum have been suspended are assumed to be weightless and inextensible, and the characteristic linear size of the disk is assumed to be small compared to the lengths of threads. In the unperturbed motion the angle the threads make with the vertical is zero, and the disk moves along the vertical and rotates around its horizontal axis. The nonlinear problem of stability of this motion is solved with respect to small deviations of the threads from the vertical. By means of canonical transformations and the Poincaré section surface method, the problem is reduced to the study of stability of the fixed point of the area-preserving mapping of the plane into itself. In the space of dimensionless parameters of the problem, regions of stability and instability are found.