Euler-Lagrange Brachistochrone Problem in rotating system

[jpo]

Homework Statement

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.

Homework 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

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?

 

[mark726]

Thank you for your question. Your approach seems to be on the right track. However, there are a few things that need to be clarified.

Firstly, in order to minimize the time of motion, the integral should be taken from \phi_0 to \phi_1, where \phi_0 is the initial angle and \phi_1 is the final angle. This will give the total time of motion.

Secondly, in the integral in the rotating frame, there should also be a term for the centripetal acceleration, which is given by r(\theta)\omega^2. So the integral should be:

T = \frac{1}{\omega} \int_{\phi_0}^{\phi_1} \sqrt{\frac{\dot{r}^2(\theta)+r^2(\theta) (\dot{\theta}+\omega)^2+r(\theta)\omega^2}{r(\theta)}} d\theta

This will give the correct expression for the time of motion in the rotating frame.

To find the trajectory r = r(\phi) in the stationary frame, you can use the constraint equation r = r(\theta) and the relation between \phi and \theta (as you did in your attempt) to eliminate \theta from the integral. This will give an expression for the time of motion in terms of r and its derivatives with respect to \phi. You can then minimize this expression with respect to r(\phi) using the Euler-Lagrange equation, as you suggested.

I hope this helps. Let me know if you have any further questions.