- #1

TubbaBlubba

## Homework Statement

We consider a pendulum of length L hanging along the z-axis with a mass (taken to be unity) at the end, attached to an arm of length R free to rotate about the z-axis but restricted to the xy-plane. The system is completely described by the angle of the pendulum rod with respect to the z-axis (theta), and the angle between the arm with respect to the x-axis (phi), and their velocities.

The objective is to derive the mass' equations of motion when subject to gravity and I'd prefer to use Lagrangian formalism; my problem in doing this is specifically is coupling the angular velocity vectors to reduce degrees of freedom (the angle phi disappears). I have spent a completely absurd amount of time on this that I really need for other things, attacking the problem in any number of ways.

## Homework Equations

The most important quantities appear to be the angular momentum about the z-axis, which is given by:

##A_z = \dot{\phi}(R^2 + L^2\sin^2\theta) + \dot{\theta}RL\cos\theta##, and the velocity squared:

##v^2 = \dot{\phi^2}(R^2 + L^2\sin^2\theta) + \dot{\phi}\dot{\theta}RL\cos\theta + \dot{\theta^2}L^2##

(Both of these I have painstakingly derived using cartesian coordinates and confirmed in various other ways).

## The Attempt at a Solution

The equations are nice and neat and precisely what you would expect, but solving the Euler-Lagrange equations is impossible if we cannot differentiate ##\dot{φ}##. The line of reasoning that has made the most sense to me so far is this: Any acceleration in the "theta direction" should by geometric argument cause an acceleration ##\ddot{\theta}(L/R)\cos\theta## in the phi direction, and the centripetal acceleration perpendicular to the z-axis (i.e. the force conserving angular momentum) an acceleration ##-\dot{\theta^2}(L/R)\sin\theta##, giving us ##\ddot{\phi} = \frac{L}{R}(\ddot{\theta}\cos\theta - \dot{\theta^2}\sin\theta)##, with the merciful solution ##\dot{\phi} = \frac{L}{R}(\dot{\theta}\cos\theta) + C##. While the EoM looked pretty sensible, solving it using these results resulted in one among many utter abominations. My basic test is just to "shut off" gravity and see if conserved quantities are conserved, which none of my attempts have passed so far.

What am I missing? I'm sure it's something relatively elementary and that there is a much simpler way of solving this (Total angular momentum?). I have done every single differentiation of every single quantity I have been able to think of (to exaggerate slightly), but nothing really works. Or is this reasoning correct? (In that case, it could be a mistake in my solution of the EL equations or my implementation.).

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