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?