What are the most beautiful equations in physics? Why?

One way of summarising physics is as a process of throwing away all messiness and complication until you reach the simplest possible system that exhibits the phenomenon you want. Do this right, model that core mathematically, and you may be lucky enough to come to the realms where pure, beautiful equations can be found.
In 1963 it took a non-physicist — Edward Lorenz — to perform the meta-step. What about the complications? Could we apply the same powerful process to the bits that everyone had been trying to throw out: that is, to strip away all the messiness from the messiness, until we found equations that summarised this pure “chaos”?
Lorenz did this by stripping down a description of convection of heat in the atmosphere until he got to the core of what made modelling these sort of systems so difficult. Here’s what he ended up with:
dxdt=σ(yx)
dydt=x(ρz)y
dzdt=xyβz
Where ρβ and σ are constants. That’s all.

James Gleick only needs a few lines to summarise the shock that this caused to physicists, as well as Lorenz’s character:
To the eye of a physicist, the equations looked easy. You would glance at them—many scientists did, in years to come—and say, I could solve that.
“Yes,” Lorenz said quietly, “there is a tendency to think that when you see them. There are some nonlinear terms in them, but you think there must be a way to get around them. But you just can't.”
(James Gleick, Chaos)
These three equations are the original, simplest and most beautiful expression of chaotic behaviour. Arbitrarily small differences in the starting values give rise of huge differences as the system evolves in time.
(A graph of the time evolution of Lorenz’s equations)
What’s more, there are characteristic phenomena that are associated with chaos, its onset, and its breakdown. Since 1963 physicists have been seriously studying features of chaotic systems, uncovering whole new laws governing physical behaviour across scales and across into distant realms such as economics.
It turned out that in throwing out what they saw as messy complication, physicists had been throwing away something profoundly important. And despite some hints in the work of Laplace, Lagrange and others, it took a meteorologist to find the gold in the spoil.

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