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Motivating examples and major applications

A dynamical system is a mathematical model of a system evolving in time. Most models in mathematical physics are dynamical systems. If the system has only a finite number of `state variables’, then its dynamics can be encoded in an ordinary differential equation (ODE), which expresses the time derivative of each state variable (i.e. its rate of change over time) as a function of the other state variables. For example, celestial mechanics concerns the evolution of a system of gravitationally interacting objects (e.g. stars and planets). In this case, the `state variables’ are vectors encoding the position and momentum of each object, and the ODE describe how the objects move and accelerate as they gravitationally interact. However, if the system has a very large number of state variables, then it is no longer feasible to represent it with an ODE. For example, consider the flow of heat or the propagation of compression waves through a steel bar containing 1024 iron atoms. We could model this using a 1024-dimensional ODE, where we explicitly track the vibrational motion of each iron atom. However, such a `microscopic’ model would be totally intractable. Furthermore, it isn’t necessary. The iron atoms are (mostly) immobile, and interact only with their immediate neighbours. Furthermore, nearby atoms generally have roughly the same temperature, and move in synchrony. Thus, it suffices to consider the macroscopic temperature distribution of the steel bar, or study the fluctuation of a macroscopic density field. This temperature distribution or density field can be mathematically represented as a smooth, real-valued function over some three-dimensional domain. The flow of heat or the propagation of sound can then be described as the evolution of this function over time.
We now have a dynamical system where the `state variable’ is not a finite system of vectors (as in celestial mechanics), but is instead a multivariate function.The evolution of this function is determined by its spatial geometry —e.g. the local `steepness’ and variation of the temperature gradients between warmer and cooler regions in the bar. In other words, the time derivative of the function (its rate of change over time) is determined by its spatial derivatives (which describe its slope and curvature at each point in space). An equation which relates the different derivatives of a multivariate function in this way is a partial differential equation (PDE). In particular, a PDE which describes a dynamical system is called an evolution equation. For example, the evolution equation which describes the flow of heat through a solid is called the heat equation. The equation which describes compression waves is the wave equation.
An equilibrium of a dynamical system is a state which is unchanging over time; mathematically, this means that the time-derivative is equal to zero.