- #1
jack476
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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.