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Re: Hamiltonian System with NDSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87972] Re: Hamiltonian System with NDSolve
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Mon, 21 Apr 2008 06:38:39 -0400 (EDT)
  • Organization: Uni Leipzig
  • References: <fuhfhi$ira$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de

Hi,

you need a symplectic method and NDSolve[] has implemente one
as Method ->{"SymplecticPartitionedRungeKutta", ..}

Regards
   Jens

Grandpa wrote:
> So I'm trying to solve the following Hamiltonian system using Mathematica.
> 
> solution = NDSolve[{x'[t] == 2p[t], x[0] == 2, 
> p'[t] == I*(2 + 1/2)(I*x[t])^(1 + 1/2), p[0] == 1-2*I},          {x, p}, {t,0,10}, MaxSteps -> Infinity][[1]];
> 
> I'm letting E=1, so at all points t, it should be that
> (p[t]/.solution)^2-(I*x[t]/.solution)^(2+1/2)=1.
> 
> That is the case until the function crosses a branch cut on the complex x-plane that runs from 0 to i*(Infinity).  Once the function crosses the branch cut, the system no longer preserves E and the value changes to something around 1.3.
> 
> How can I go about fixing this problem?  I've already tried working with the precision and accuracy goals, and that didn't work.  I'm not sure what else to consider.  Any help would be appreciated.
> 
> Thanks,
> Alex
> 


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