Why is this system not Hamiltonian?

Question

I have a three dimensional dynamical system

$$ \dot x =f(x,y,z)= -y \\ \dot y =g(x,y,z)= x \\ \dot z =h(x,y,z)= -x^2-y^2$$

and I am asked to say why this isn't Hamiltonian.

Now I know for a system to be Hamiltonian it must be a first integral and preserve volume.

This system does preserve volume as $$\frac{df(x,y,z)}{dx} + \frac{dg(x,y,z)}{dy} +\frac{dh(x,y,z)}{dz} =0$$

I am really confused by this.

Answer 1

In general a system with $2N$ variables $({\bf x}, {\bf y})$ is said to be Hamiltonian if there exists a function $H = H({\bf x}, {\bf y}, t)$ such that

\begin{eqnarray} \frac{{\rm d}{\bf x}}{{\rm d}t} &=& +\frac{\partial H}{\partial {\bf y}} \\ \frac{{\rm d}{\bf y}}{{\rm d}t} &=& -\frac{\partial H}{\partial {\bf x}} \end{eqnarray}

You system has 3 coordinates, so it cannot be written this way.