Finding energy as a function of Symplectic area?

[BiotFartLaw]

Homework Statement

Find the energy E of the harmonic oscillator (H(x,p)=p²/2m+mω²x²) as a function of the system's symplectic area.

Homework Equations

Canonical equations and A=$\int p dx$ (over one period)

The Attempt at a Solution

From Hamilton's equations I get :

$\dot{x}=\partial H/ \partial p$ and $\dot{p}=- \partial H/ \partial x$

So

$dot{x}=p/m$ and $\dot{p}=-2m\omega²x$

$x(t)=pt/m ; p(t)=-2m \omega ²xt$

Then I integrate

$\int pdx = \int p d(pt/m)$

But I'm not sure how to handle the d(pt/m) term. If I do chain rule (in time) I get something like

$d(pt/m)=1/m (p+\dot{p}))dt$

and I'm not really sure what the answer is if I do it in x. Since I don't know what dp/dx is. (other than m*dv/dx=m*dx'/dx ... but I'm not sure what good that does me).

 

[TSny]

Think about the shape of the orbit (for a given energy E) on a p vs x plot. There is a simple geometric formula for the area enclosed in the orbit,$\int{pdx}$.

[EDIT: Or, if you really want to carry out the integration, use E = p²/2m + (1/2)mω²x² to find p as a function of x for fixed E.]