# Symplectic integrator/hamiltonian

1. May 9, 2007

### nlake27

I'm trying to get a numerical solution for a hamiltonian mechanics problem. According to wikipedia, there's a method of solving the resulting differential equations called a symplectic integrator that's designed specifically for such problems, but my computational physics textbook doesn't mention it. If anyone could point me in the direction of a good basic resource (website, text, whatever), I'd be grateful. Thanks.

2. May 9, 2007

### lalbatros

Have you read this on wiki: http://en.wikipedia.org/wiki/Symplectic_integrator
It is rather clear that you have to represent equation (2) or (3) in some way.
This will end-up, I think, to solving linear systems of equations on short time steps.
In other words, any method will do it.
My preffered is based on diagonalisation, just as in QM. It may not be the fastest.
What I remember the best from QM is how to compute practically the exponential of an operator (by diagonalisation).

3. May 9, 2007

### nlake27

Yes, I've read this; I'm looking for something more detailed. The Hamiltonian in question yields a dimension-2 system of messy nonlinear differential equations that are, I think, non-separable. The numerical solutions need to be pretty precise because it's a chaotic system.

4. May 9, 2007

### lalbatros

Numerically, I think, it doesn't make a difference if it is a chaotic system or not.
But you need to choose a good method, of course.
I suggest you to try first by integrating some other system with possibly an analytical solution to check for errors.
I also suggest you to use existing librairies: IMSL, NAG, see netlib, and many othes, look also for matlab
Try to find a software that provides error estimations, possibly.
If not possible, try to look at the statbility of your solution with respect to the time-step or other parameters used for improving the precision.
Consider adaptative method and implicit methods.

Keywords: numerical integration of differential equations