From: nebusj@rpi.edu (Joseph Nebus)
Subject: Re: meta-stability
Date: 1 Jun 2001 06:59:55 -0400
Newsgroups: sci.math.num-analysis,sci.physics
Summary: Symplectic integrators (numerical integration respecting conservation laws)

John Creighton <JohnCreighton_@hotmail.com> writes:

>Pierre Asselin wrote:

>> If you can put your equations in Hamiltonian form, you can use a
>> symplectic integrator.  I did a quick search on netlib and came up
>> with nothing :-(  But it's clearly the way to go.  Head for the
>> library, I guess.

>I am not sure how this would work. Wouldn't each iteration of the integrator
>cause an error and on average after n iterations we would find the energy of
>the system has increased by a*n^p. Consider a ball immersed in a friction
>less fluid. On average as the number of iterations goes to infinity the
>magnitude of the angular velocity will get faster and faster. This is
>clearly undesirable when modeling a system.


	Symplectic integrators are magic.  That overstates things, but 
really only very slightly.  They're designed for Hamiltonian systems, 
which have nice properties like conservation of energy and such, and 
so make use of those conservation rules.  

	There will be errors, but they tend not to be the ones that 
mess up the dynamics of the system.  As one person teaching them to me 
explained, let's say you have an integrator modeling the orbit of the 
earth for one billion years into the future.  Your nonsymplectic 
integrator may put the Earth one A.U. off its correct location, in the 
center of the sun.  Your symplectic one, though, if it has made an 
error of one A.U. in its position, will put it one radian ahead (or 
behind) in its orbit, but still in its correct orbit.  Given the choice 
between mistakenly having an early spring and mistakenly having all 
life on earth extinguished in a firely blaze, most people prefer the 
first.

	A great reference for this, explained completely enough that 
you can write software directly from it, is "Symplectic Integration 
of Hamiltonian Systems," by PJ Channell and C Scovel, writing in the 
journal Nonlinearity 3 (1990), pages 231 to 259.  Includes examples 
of how very well it conserves energy for unspeakable billions of 
timesteps.  A two-minute search of the web doesn't turn up a copy of 
it online, but any university library should either have a copy or be 
able to get one.

								Joseph Nebus
------------------------------------------------------------------------------