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*From*: John Denker <jsd@av8n.com>*Date*: Mon, 31 Dec 2012 03:22:31 -0700

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

involved.

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.

http://www.av8n.com/physics/symplectic-integrator.htm

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