I wonder about showing conservative, hamiltonian systems cannot have any attracting fixed points.
For example
Definition. Let , be a smooth real-valued function of two variables. Then a system of the form
$\dot{x}=\frac {\partial H}{\partial y}$
$\dot{y}=-\frac {\partial H}{\partial x}$
is called a Hamiltonian system and the function H is called the Hamiltonian. (a) Show that for any Hamiltonian system , $H(x,y)$ is a conserved quantity. (b) Show that Hamiltonian systems cannot have any attracting fixed points. (c) Convert the equation $m\ddot{x} + kx = 0$ into a dynamical system and construct the associated Hamiltonian function H. ,
first thing is
$\frac {dH(x,y)}{dt} = \frac {\partial H}{\partial x}*\frac {\partial x}{\partial t} + \frac {\partial H}{\partial y}*\frac {\partial y}{\partial t}=-\dot{y}\dot{x}+\dot{x}\dot{y}$
when i show the "cannot attracting fixed points" i made $\frac {\dot{y}}{\dot{x}}$ and found it equals $-\frac {\partial y}{\partial x}$ but i cant make clear it