Chapter 3 - Mathematical models

Summary

Most of the physical processes illustrated in the previous chapter are conveniently described by a set of partial differential equations (PDEs) for a vector field Ψ(x, t) which represents the state of the system in phase space X. The coordinates of Ψ are the values of observables measured at position x and time t: the corresponding field theory involves an infinity of degrees of freedom and is, in general, nonlinear.

A fundamental distinction must be made between conservative and dissipative systems: in the former, volumes in phase space are left invariant by the flow; in the latter, they contract to lower dimensional sets, thus suggesting that fewer variables may be sufficient to describe the asymptotic dynamics. Although this is often the case, it is by no means true that a dissipative model can be reduced to a conservative one acting in a lower-dimensional space, since the asymptotic trajectories may wander in the whole phase space without filling it (see, e.g., the definition of a fractal measure in Chapter 5).

A system is conceptually simple if its evolution can be reduced to the superposition of independent oscillations. This is the integrable case, in which a suitable nonlinear coordinate change permits expression of the equations of motion as a system of oscillators each having its own frequency.