Coupled Generalized Momentum & Hamiltonian Mechanics

[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?

 

[Jhero]

Thank you for sharing your brilliantly engineered system with us. I can understand your confusion about the coupled generalized momenta in your Lagrangian. However, I believe there is a way to solve this problem.

Firstly, I suggest rewriting your Lagrangian in terms of the generalized coordinates x and \theta instead of the velocities \dot{x} and \dot{\theta}. This will give you a Lagrangian of the form L=L(x,\theta,\dot{x},\dot{\theta}). Then, you can use the Euler-Lagrange equations to obtain the equations of motion for x and \theta.

Next, you can use the equations of motion to eliminate the velocities \dot{x} and \dot{\theta} from your expressions for the generalized momenta p_x and p_\theta. This will give you a set of equations that relate the generalized momenta to the generalized coordinates and their time derivatives.

Finally, you can use these equations to solve for the generalized momenta in terms of the generalized coordinates and their time derivatives. This will give you a set of equations that relate the momenta to the coordinates, which you can then use to solve for the motion of your system.

I hope this helps you in solving your problem. Let me know if you have any further questions or if you need any more clarification.