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Finding energy as a function of Symplectic area?

  1. Mar 12, 2013 #1
    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=[itex]\int p dx[/itex] (over one period)


    3. 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 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. jcsd
  3. Mar 12, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    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
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