jdwood983
Oct29-09, 07:53 PM
I have a brilliantly engineered system of a bead-on-a-circular-loop (mass=m) rigidly attached to a massive block (mass=M) on one side and a spring on the other. The spring motion is constrained to be in x-direction only, while the bead is free to move on the wire anyway it wants to (no \phi dependence though). Simple picture looks like:
|\/\/\/\/\/-O-[]
With O being the bead-on-a-loop part, \/\/\/\/ is the spring (spring constant=k), and [] is the block. I set x to be the from the left wall to the center of the loop and the angle \theta to be the counter-clockwise angle from the +x axis. In doing this, I got the Lagrangian to be:
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
but when I take my Legendre transform,
p_{x}=\frac{\partial L}{\partial\dot{x}}=(M+m)\dot{x}-mr\dot{\theta}\sin\theta
p_{\theta}=\frac{\partial L}{\partial\dot{\theta}}=r^{2}\dot{\theta}-mr\dot{x}\sin\theta
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?
|\/\/\/\/\/-O-[]
With O being the bead-on-a-loop part, \/\/\/\/ is the spring (spring constant=k), and [] is the block. I set x to be the from the left wall to the center of the loop and the angle \theta to be the counter-clockwise angle from the +x axis. In doing this, I got the Lagrangian to be:
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
but when I take my Legendre transform,
p_{x}=\frac{\partial L}{\partial\dot{x}}=(M+m)\dot{x}-mr\dot{\theta}\sin\theta
p_{\theta}=\frac{\partial L}{\partial\dot{\theta}}=r^{2}\dot{\theta}-mr\dot{x}\sin\theta
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?