# Euler-Lagrange Brachistochrone Problem in rotating system

1. Aug 9, 2011

### jpo

1. The problem statement, all variables and given/known data
Bead slides on a wire (no friction) shaped as $r = r(\theta)$ in the Oxy plane. The Oxy plane and the constraining wire rotate about Oz with $\omega = const$
$r, \theta$ is the rotating polar frame; $r, \phi$ is the stationary frame.
Find the trajectory $r = r(\phi)$ in the stationary frame.

2. Relevant equations
Minimize the time of motion
$T = \frac{1}{\omega} \int_{\pi/2}^{\phi_1} \sqrt{\frac{r'^2+r^2}{r}} d\phi$
in polar coordinates

3. The attempt at a solution
The trajectory $r = r(\phi)$ in the inertial frame is unknown; In the rotating frame the trajectory is known: it is $r = r(\theta)$. Then the integral in the rotating frame becomes

$T = \frac{1}{\omega} \int_{0}^{t_1} \sqrt{\frac{\dot{r}^2(\theta)+r^2(\theta) (\dot{\theta}+\omega)^2}{r(\theta)}} dt$,
because
$\phi=\theta+\omega t$ and
$\frac{dr}{d\phi}=\frac{\dot{r}}{\dot{\phi}}=\frac{\dot{\theta}}{\dot{\theta}+\omega}$

I have tried to write the integral into the rotating polar frame and then enforce the constraint, by substituting $r$ and its derivative through $\theta$ and $\dot{\theta}$. Then I'll minimize $T$with Euler-Lagrange
$\frac{\partial J}{\partial\theta}-\frac{d}{dt}\frac{\partial J}{\partial\dot{\theta}}=0$

Is this correct?