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← 02-28 Symplectic Solver

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Showing Revision 2 created 10/24/2012 by Amara Bot.

  1. Another one to take the forward Euler's method that you already know so well
  2. and turned it into one of these wondrous methods that consist of phase space area.
  3. Here I wrote down the equations for standard forward Euler's method.
  4. Now, there is a single change that's going to do the job.
  5. I do not compute the force at the original position. I compute the force at the new position.
  6. This leads to the following. This looks like a programming error on first sight.
  7. Soon, I'll be using the initial values on the right-hand side to compute the new values,
  8. but actually, it turns out that this the right thing to be doing in contrast to the forward Euler's method.
  9. Now, these two steps are executed in sequence. First, we change the position.
  10. And you already saw that this does not change
  11. the area in phase space and then we take the new position.
  12. The velocity remained unchanged and compute the new velocity,
  13. which also does not change the area in phase space, which we saw before
  14. and then we repeat the first step and so on and so on.
  15. This is now a symplectic method. It does not change the area in phase space.