I am given the following Lagrangian:
$$L = \frac{ml^2}{2}\left((\dot\theta)^2 + \sin^2{(\theta)}\dot\phi^2\right) + \frac{I\dot\phi^2}{2} + mgl\cos{\theta}$$
which is meant to represent a simple pendulum mounted on a rotating turntable, and $m, g, l$, and $I$ are some physical constants. The task is to obtain the equations of motion, which I think I did, but we were given no solutions, so I can't know for sure if I am correct or not.
To solve it I applied the Euler-Lagrange equation (ELE), separately on both the variables $\theta$ and $\phi$:
0 = $\frac{\partial L}{\partial \theta} - \frac{d}{dt}\left(\frac{\partial L}{\partial\dot\theta} \right) = \frac{ml^2}{2}\dot\phi^2\sin{2\theta}-mgl\sin{\theta} + ml^2 \ddot\theta$
from which the first equation of motion is: $\ddot\theta = \frac{g}{l}\sin{\theta}-\frac{1}{2}\dot\phi^2\sin{\theta}$.
Then I applied the ELE for $\phi$: 0 = $\frac{\partial L}{\partial \phi} - \frac{d}{dt}\left(\frac{\partial L}{\partial\dot\phi} \right) = (-ml^2\sin^2{\theta}-I)\ddot\phi = 0$.
This is where I sort of ran into trouble. My next reasoning was that since the term in the brackets isn't always zero, the only way for the equation to be satisfied is if $\ddot\phi=0$, but I don't know if this is appropriate in Lagrangian mechanics.