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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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