1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]


    [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


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted