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Re: Hamiltonian System

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88103] Re: Hamiltonian System
  • From: dh <dh at metrohm.ch>
  • Date: Thu, 24 Apr 2008 05:59:53 -0400 (EDT)
  • References: <fumquk$sdm$1@smc.vnet.net>


Hi Alex,

you have branch cuts in the differential equation as well as in the 

energy expression.

To deal with the first branch cut in the DE we may define a power 

function with a different branch cut: E.g. To integrate over the first 

branch cut at approx. t=2, we may define

myPow[x_]:= (Exp[-I Pi/4]Sqrt[Exp[+I Pi/2]I  x])^3

this has a branch cut along the line trough I instead of -1. This line 

is not cut by I x[t] if we integrate from 0..3. We may therefore use 

myPow  in place of (I x[t])^(1+1/2).

With myPow we get a smooth curve, but still the energy is not constant 

near t=2. This comes from the energy term: (I*x[t]/.solution)^(2+1/2) 

that has a cut near t=2.



hope this helps, Daniel



Grandpa wrote:

> So I'm trying to solve the following Hamiltonian system using Mathematica.  I posted this earlier, but I wanted to make sure it's not skipped over.

> 

> 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 also tried the SymplecticPartitionedRungeKutta method, but that didn't work either.  I'm not sure what else to consider. Any help would be appreciated.

> 

> Thanks,

> Alex

> 




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