On a Qualitative and Lie Symmetry Analysis for a Pendulum with Two Reaction Wheels

Abstract

In this article, it is studied the mechanical system formed by a pendulum with two reaction wheels in which the friction torque is assumed to follow a Coulomb law. A qualitative analysis of the system is performed for the damped case. Specifically, the equilibrium points for the unforced pendulum are analyzed. Also, in the forced case, the conditions for which there exist asymptotically stable solutions are determined. In order to study the exact analytical solution of the unforced pendulum, we also perform a Lie symmetry analysis. In this regard, it is shown that the exact general solution of the system for null motor torques can be expressed in terms of the general solution to an Abel equation. In the unforced and undamped case, the exact general solution is obtained in explicit form and expressed in terms of the Jacobi elliptic function by using the Lie symmetry approach.