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[Phys-L] visualizing the phase space of a simplectic integrator

Hi -

For a wide range of numerical integration problems, especially of the sort
that show up in physics, a symplectic integrator works dramatically better
than the other kind. I decided to write up my notes on the basics principles

I find that if I understand an equation, I can draw a picture of what's
going on ... and vice versa ... so I cobbled up a bunch of pictures.
They show the region of phase space delimited by an ensemble of systems,
and show how the region evolves. The symplectic update rule has the
nice feature that it is guaranteed to preserve the area of such a region.

This is related to Liouville's theorem, which is related to the second
law of thermodynamics and (not coincidentally) to the Heisenberg uncertainty
principle, the optical brightness theorem, the fluctuation/dissipation
theorem, et cetera.

A first-order symplectic Euler integrator is not one iota harder to
implement than an old-fashioned forward Euler integrator, and it works
vastly better.