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Part I - Motivating examples and major applications

Published online by Cambridge University Press:  05 June 2012

Marcus Pivato
Affiliation:
Trent University, Peterborough, Ontario
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Summary

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 describes how the objects move and accelerate as they interact gravitationally.

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 is not 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.

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Publisher: Cambridge University Press
Print publication year: 2010

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