# Finding energy as a function of Symplectic area?

1. Mar 12, 2013

### BiotFartLaw

1. The problem statement, all variables and given/known data
Find the energy E of the harmonic oscillator (H(x,p)=p2/2m+mω2x2) as a function of the system's symplectic area.

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

3. 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 dunno what dp/dx is. (other than m*dv/dx=m*dx'/dx ... but I'm not sure what good that does me).

Last edited: Mar 12, 2013
2. Mar 12, 2013

### 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)2x2 to find p as a function of x for fixed E.]

Last edited: Mar 12, 2013