
Hamiltonian Equation
Just wondering if you have
dx/dt = y^3  y
dy/dt = x
then you can find the Hamiltonian which is H(x,y) = (x^2)/2 +(y^4)/2 (y^2)/2
My question is what does the hamiltonian say about the nature of the steady states i know that (0,0) is a saddle and (0,1) is a centre
My second question is how can you describe the nature of the critical points of H(x,y). Im assuming it might help to draw a phase plane
thanks

I believe the Hamiltonian is actually $\displaystyle x^2/2+y^2/2y4/4$. The steady states (i.e. critical points) occur at (x,y) = (0,0), (0,1), and (0,1). Isn't (0,0) the center and (0,1) and (0,1) the saddles?
$\displaystyle H = x^2/2 + V(y), with V(y) =y^2/2y^4/4$
The potential function, $\displaystyle V(y)$ has the following properties:
$\displaystyle V'(y) = 2y y^3$
$\displaystyle V''(y) = 2  3y^2$
Therefore V(0) is a minimum, and hence a stable point in the system, and V(1) and V(1) are maxima in the potential function, and hence unstable.
Not sure this is getting at your question...