# Coupled Generalized Momentum & Hamiltonian Mechanics

1. Oct 29, 2009

### jdwood983

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?