# Homework Help: Construct the Lagrangian for the system

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1. Dec 7, 2017

### TSny

If the only freedom of motion is rotation of the whole system about the central vertical axis, then I can't see anything interesting happening.

My first interpretation was to assume the upper rod is fixed, but that the wires and lower rod can undergo "twisting" motion such that $\varphi$ is the angle of twist about a vertical axis through the center of the rod. The only motion of the center of the rod would be vertical motion where the height of the center of the rod would be a function of $\varphi$. It's not too hard to set up the Lagrangian for this case, but getting the equation of motion from the Lagrangian is a bit cumbersome (unless you assume small twisting oscillations so that you can use a small angle approximation). Likewise, the expression for the quadrature is messy. So, I didn't think that this interpretation would make a very good exam question. But, of course, who am I to make that decision?

2. Dec 7, 2017

### proton4ik

Your interpretation seems reasonable. Wouldn't Lagrangian look the same as I wrote in the question considering this interpretation?

3. Dec 7, 2017

### TSny

To what point do the coordinates $x$, $y$, and $z$ refer in your equations?

4. Dec 7, 2017

### proton4ik

To the left end of the rod

5. Dec 7, 2017

### TSny

If the rod rotates about a vertical axis through its center, then the coordinates of the left end of the rod would not be given by your equations for $x$, $y$, and $z$. Also, the kinetic energy of the rod would not be given by 1/2 the mass times the square of the speed of the left end. There would be rotational kinetic energy plus translational kinetic energy due to vertical motion of the center of mass of the rod.