The Double Pendulum

Most of you are familiar with the simple pendulum: A small heavy object suspended at the end of a thin light string and set into oscillation. For small oscillations (and in the absence of air friction) the motion is that of a "harmonic oscillator": That is, periodic motion with a period proportional to the sqaure-root of the length of the pendulum. For larger oscillations, the motion of the simple pendulum is still periodic but no longer given by a simple formula. Indeed, for large oscillations the equations governing the motion fo the pendulum are non-linear in contrast to the linear equations for small oscillations. However while the equations are nonlinear, the motion is still regular and predictable. A double pendulum consists of two simple pendulums in tandem: One attaches a single pendulum to the end of another! The equations of otion are again non-linear for large oscillations but now the motion becomes quite irregular and very sensitive to the initial conditions. This kind of behaviour is the hallmark of chaos. See the simulation of the double pendulum in the references. Chaos occurs in many nonlinear systems and it implies that even systems with a few degrees of freedom, and hence naively simle, can show complicated behaviour which is essentially unpredictable on long time scales. However chaos is very different from randomnes: The former arises in perfectly deterministic systems while the later is intrinsically nondeterministic, and the distinction between the two at the prcatical level can be seen by looking at the "phase space" of the system, as we shall see later.