Dynamic and Static Stability
The motion of dynamical systems is strongly dependent on whether their behavior is stable or unstable. This chapter obtains the conditions which determine stability for certain types of systems that we have analyzed in previous chapters, using a direct method that involves the properties of the potential energy or the dynamic potential energy. We also discuss an indirect method that first requires the linearization of the equations of motion, but which is applicable to a wider class of systems. Since the direct dynamic stability method relies on the behavior of the potential energy, a quantity that in most dynamical systems is independent of time, this stability criterion is also applicable to static systems, i.e., systems that are designed to inherently be in equilibrium. Stability of static systems is rarely treated in any depth in traditional statics or dynamics courses so we will also analyze some of the important ways in which such static systems can lose stability.