A particle on the edge of an inclined spinning disc

[jack476]

Homework Statement

A particle is fixed to the edge of a disc of negligible mass making an angle θ₀ with the ground in a uniform gravitational field, and is free to rotate about the center of the disc. I need to find the equations of motion.

Homework Equations

The definition of the Lagrangian in spherical coordinates,

$\frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2sin^2{\theta}\dot{\phi}^2) - U(r, \phi, \theta)$

where $\theta$ is given to be the angle of altitude and $\phi$ is the azimuthal angle.

The Attempt at a Solution

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It was easy to find the Lagrangian to be

$L = \frac{R^2}{2}m(\dot{\theta}^2 + sin^2{\theta}\dot{\phi}^2) + mgrcos{\theta}$

It is from here that I do not know how to proceed. Wouldn't using that Lagrangian to find the equations of motion for $\phi$ and $\theta$ just get me the equations of motion for a particle constrained to move?

The first thing I've thought to try is to treat the particle as though it is moving on the surface of a sphere, fixed to move along the great circle produced by the plane passing through the center of the sphere and making an angle θ₀ with its equator, then I could use the equation of the great circle as a constraint relating $\phi$ and $\theta$, but I'm struggling to find the parametric equations for a great circle.

The other thing I tried was to use cylindrical coordinates and impose the constraint that the height $z$ is some function of $\phi$. My guess is that this function would be something of the form $z = Rcos{\theta_0}cos{\phi}$, then use $f = z - Rcos{\theta_0}cos{\phi} = 0$ as my equation of constraint. This gets me the equations

$\ddot{\phi} + \frac{\lambda}{R}cos{\theta_0}sin{\phi} = 0$
$\ddot{z} - g + \frac{\lambda}{m} = 0$

It is from there that I do not know how to proceed.

 

[theodoros.mihos]

How you write the Lagrangian for horizontal disk on external field by angle $\theta_0$?
This Lagrangian produce the same equation of motion? Must be?

 

[haruspex]

Are you taking the z axis as vertical? Doesn't that make phi and its derivatives a bit tricky?
Have you tried using cylindrical coordinates but taking the z axis perpendicular to the disk?