Lesson plan
Phase space method
In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is transformed back into the original setting. The phase space method is used widely in physics. It can be applied, for example, to find traveling wave solutions of reaction–diffusion systems.
Article summary
In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is transformed back into the original setting. The phase space method is used widely in physics. It can be applied, for example, to find traveling wave solutions of reaction–diffusion systems.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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