In this chapter we discuss the general problem of the stability of a dynamical system. The problem can be studied by first finding the conditions for the system to be in a state of equilibrium — when all the forces acting on the system balance to zero — and then determining whether the equilibrium is stable or unstable. Near the equilibrium state the system can be described approximately as an ensemble of coupled oscillators, and we can find the characteristic frequencies of oscillations and determine whether the motion of the oscillators is bounded or unbounded. In the first case, we say that the system is in a stable equilibrium state and in the second that is in an unstable equilibrium state. The characteristic frequencies also define the response of the system to additional external forces. This method of studying the stability of a dynamical system and its motion near an equilibrium point is very general, and can be used to study very different dynamical systems, ranging from musical instruments, to a building subject to an earthquake, to molecules, or even to a simple pendulum.
KeywordsEquilibrium Point Stable Oscillation Simple Pendulum Stable Equilibrium State Unstable Point
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