Dimensional Analysis, Chaos and Self-Organised Criticality

Introduction The increasingly powerful mathematical tools described in Chapter 7 provided the means for tackling complex dynamical problems in classical physics. Despite these successes, in many areas of physics problems can become rapidly very complex and, although we may be able to write down the differential or integral equations which describe the behaviour of the system, often it is not possible to find analytic solutions. The objective of this chapter is to study techniques developed to tackle these complex problems, some of them so non-linear that they seem quite beyond the scope of traditional analysis. First, we review the techniques of dimensional analysis . Used with care and insight, this approach is powerful and finds many applications in pure and applied physics. We will give as examples the non-linear pendulum, fluid flow, explosions, turbulence and so on. Next, we briefly study chaos , the analysis of which became feasible only with the development of high-speed computers. The equations of motion are deterministic and yet the outcome is extremely sensitive to the precise initial conditions. Beyond these examples are even more extreme systems, in which so many non-linear effects come into play that it is impossible to predict the outcome of an experiment, in any conventional sense. And yet regularities are found in the form of scaling laws. There must be some underlying simplicity in the way in which the system behaves, despite the horrifying complexity of the many processes involved. These topics involve fractals and the burgeoning field of self-organised criticality .