Solve this Hamiltonian System in Several Ways

[ResiRadloff]

Homework Statement

Let us denote by $\textbf{z} = (x,y) \in \mathbb{R}^2$ the Cartesian coordinates of a point in the plane.

1. Given the Hamiltonian $H(\textbf{z}) = \frac{|\textbf{z}|^2}{2}$, write down the corresponding canonical Hamiltonian system for $\textbf{z}(t)$.

2. Write down the analytical solution of this system.

In the following, assume that the system reads $\frac{d}{dt}\textbf{z} = \textbf{F}((\textbf{z}(t)))$ where $\textbf{F} = (F_x,F_y)$, and let $t^n = n\Delta t$ where $\Delta t > 0$ is the time step size. Also let $\textbf{z}^n = \textbf{z}(t^n)$.

3. Solve the Hamiltonian system numerically by using the following numerical methods. For each of them write down the scheme, plot your result vs. the analytical result, and plot energy (= Hamiltonian) and $L^2$-error of the scheme. Start with a random point $(x_0,y_0) \in [0,1]^2$ at $t = 0$ and evolve $\textbf{x}(t)$ forward in time. You may use $\Omega = 2, \Delta t = 0.03$, and do 500 time steps.

(a) The explicit Euler method: $\textbf{z}^{n+1} = \textbf{z}^n + \Delta t \textbf{F}(\textbf{z}^n) ; \textbf{z}^0 = \textbf{z}(t=0)$

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Relevant Equations

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Good evening,
unfortunately I can't get to the solution of my task

I wrote for the system:

$\frac{dz}{dt} = \nabla_p H ; \\ \frac{dp}{dt} = - \nabla_z H$

Then the solution would be (as $\nabla_p H = 0)$:

$\frac{dz}{dt} = 0 \Rightarrow z = const.$ and $p = zt + p_0$.

But that can't be as now finding a numerical solution doesn't make sense?.

I would be really happy if someone yould help me?

Thanks a lot
Resi

Danke schonmal und viele Grüße

 

[pasmith]

Are you not looking at $$\begin{split} \frac{dx}{dt} &= \frac{\partial H}{\partial y} \\ \frac{dy}{dt} &= -\frac{\partial H}{\partial x}\end{split}$$ That is how I would interpret "hamiltonian System" if given a "hamilonian" which is a function of exactly two variables.

 

[ResiRadloff]

Okay, thanks, yeah that makes more sense.

Does anyone know how to calculate the Energy-error and what is meant by $L^2$ error?