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Hamilton-Jacobi problem

  1. Feb 18, 2012 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Using Hamilton-Jacobi's equation, find the motion equations and the trajectory of a particle in the field [itex]U (\vec r )=\frac{m \omega}{2} (x^2+y^2)[/itex].


    2. Relevant equations
    [itex]H(q_1,...q_s,\frac{\partial S _0}{\partial q_1},...,\frac{\partial S _0}{\partial q_s})=E[/itex].
    Where [itex]S_0[/itex] is the abbreviated action, apparently worth [itex]\int \sum _i p_i dq_i[/itex].


    3. The attempt at a solution
    I'm self studying CM for a final exam on 7th of March (I can choose not to take the exam if I feel not ready but this would be a pain for the next semester). I've searched google and this forum for similar problems but didn't find anything that could really help me. What I found was heavy abstract math rather than applied problems like this one.
    Ok my attempt: I don't know if I can assume that the motion is in 3 dimensions or 2. Let's take 2 for the sake of simplicity.
    I use Cartesian coordinates so that the velocity is [itex]\dot {\vec r }=\dot x \hat i + \dot y \hat j[/itex]. Thus the total energy of the system is worth [itex]\frac{m (\dot x ^2 + \dot y ^2 )}{2}+\frac{m \omega}{2} (x^2+y^2)[/itex].
    The Hamiltonian is then [itex]H=E=\frac{1}{2m} (p_x ^2+ p_y ^2)+\frac{m \omega }{2}(x^2+y^2)[/itex].
    This is where I'm stuck. It looks like I must express the generalized momenta into the abbreviated actions, but I don't know how to do so. Any tip is greatly appreciated!
     
  2. jcsd
  3. Feb 19, 2012 #2
    use the poisson brackets
     
  4. Feb 19, 2012 #3

    fluidistic

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    Gold Member

    Thanks for your reply.
    Somehow I don't see how this can help me. I'd appreciate if you could specify a bit more.
    I've checked into http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation (there are several examples), in Landau's book and in Goldstein's book. None seems to involve any Poisson bracket.
    I am not 100% sure, but it seems that from [itex]E=\frac{1}{2m} (p_x ^2+ p_y ^2)+\frac{m \omega }{2}(x^2+y^2)[/itex] I can replace [itex]p_i[/itex] by [itex]\frac{\partial S _0 }{\partial q_i}[/itex]. I doubt if it shouldn't be [itex]\frac{\partial S }{\partial q_i}[/itex] or [itex]\frac{\partial S _i }{\partial q_i}[/itex] instead.


    This would make [itex]\frac{1}{2m} \left [ \left ( \frac{\partial S _0}{\partial x} \right )^2+\left ( \frac{\partial S _0}{\partial y} \right )^2 \right ] +\frac{m\omega (x^2+y^2) }{2}=E[/itex]. I don't know what to do next. Should I solve for S?

    Wikipedia takes the example of a particle in spherical coordinates. Laudau threats the same problem. However Landau totally depreciated a term in the potential function "because it's not interesting physically" or something like that. Wikipedia however do not simply get rid of a term because it's of few interest. I really don't understand what to do next...
     
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