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Coupled Generalized Momentum & Hamiltonian Mechanics

  1. Oct 29, 2009 #1
    I have a brilliantly engineered system of a bead-on-a-circular-loop (mass=[itex]m[/itex]) rigidly attached to a massive block (mass=[itex]M[/itex]) on one side and a spring on the other. The spring motion is constrained to be in [itex]x[/itex]-direction only, while the bead is free to move on the wire anyway it wants to (no [itex]\phi[/itex] dependence though). Simple picture looks like:

    |\/\/\/\/\/-O-[]

    With O being the bead-on-a-loop part, \/\/\/\/ is the spring (spring constant=[itex]k[/itex]), and [] is the block. I set [itex]x[/itex] to be the from the left wall to the center of the loop and the angle [itex]\theta[/itex] to be the counter-clockwise angle from the [itex]+x[/itex] axis. In doing this, I got the Lagrangian to be:

    [tex]
    L=\frac{1}{2}\left(M+m\right)\dot{x}^{2}+\frac{1}{2}mr^{2}\dot{\theta}^{2}-mr\dot{x}\dot{\theta}\sin\theta-\frac{1}{2}kx^{2}-mgr\sin\theta
    [/tex]

    but when I take my Legendre transform,

    [tex]
    p_{x}=\frac{\partial L}{\partial\dot{x}}=(M+m)\dot{x}-mr\dot{\theta}\sin\theta
    [/tex]

    [tex]
    p_{\theta}=\frac{\partial L}{\partial\dot{\theta}}=r^{2}\dot{\theta}-mr\dot{x}\sin\theta
    [/tex]

    I have never seen a problem with a coupled generalized momenta like this and am stuck here. I tried solving it in a linear equation, but it kept looping through like a thousand times (okay, I didn't really go that far, but after the second time you see that you'll endlessly repeat yourself).

    I am not sure what to do, any suggestions?
     
  2. jcsd
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