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BiotFartLaw
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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=[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\omega2x[/itex]
[itex]x(t)=pt/m ; p(t)=-2m \omega 2xt[/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 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).
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