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## Homework Statement

Find the energy E of the harmonic oscillator (H(x,p)=p

^{2}/2m+mω

^{2}x

^{2}) as a function of the system's symplectic area.

## Homework Equations

Canonical equations and A=[itex]\int p dx[/itex] (over one period)

## The Attempt at a Solution

From Hamilton's equations I get :

[itex]\dot{x}=\partial H/ \partial p[/itex] and [itex]\dot{p}=- \partial H/ \partial x[/itex]

So

[itex]dot{x}=p/m[/itex] and [itex]\dot{p}=-2m\omega

^{2}x[/itex]

[itex]x(t)=pt/m ; p(t)=-2m \omega

^{2}xt[/itex]

Then I integrate

[itex]\int pdx = \int p d(pt/m)[/itex]

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

[itex]d(pt/m)=1/m (p+\dot{p}))dt [/itex]

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).

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