Sphere Surface Dynamics - Bird's Nest

This sort of problem lends itself perfectly to a Lagrangian Mechanics solution. Define the spatial coordinates (I'm going to drop the parentheses for the trig arguments):

$x = cos \theta \, cos \phi \\ y = sin \theta \, cos \phi \\ z = sin \phi$

Derive the $xyz$ velocities:

$\dot{x} = cos \theta (-sin \phi \, \dot{\phi}) + cos \phi (-sin \theta \, \dot{\theta}) \\ \dot{y} = sin \theta (-sin \phi \, \dot{\phi}) + cos \phi (cos \theta \, \dot{\theta}) \\ \dot{z} = cos \phi \,\dot{\phi}$

Particle kinetic energy:

$E = \frac{1}{2} m v^2 = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2 ) = \frac{1}{2} m \Big(\dot{\phi}^2 + cos^2 \phi \, \dot{\theta}^2 \Big)$

Particle potential energy:

$U = m g z = m g \, sin \phi$

Lagrangian:

$L = E - U = \frac{1}{2} m \Big(\dot{\phi}^2 + cos^2 \phi \, \dot{\theta}^2 \Big) - m g \, sin \phi$

Equations of Motion:

$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{\theta}}} = \frac{\partial{L}}{\partial{\theta}} \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{\phi}}} = \frac{\partial{L}}{\partial{\phi}}$

Results of evaluating equations of motion:

$\ddot{\phi} = -sin \phi \, cos \phi \, \dot{\theta}^2 - g \, cos \phi \\ \ddot{\theta} = 2 \, tan \phi \, \dot{\theta} \, \dot{\phi}$

Numerically integrating for 30 seconds and plotting yields the beautiful "bird's nest" pattern seen in the problem statement. At $t = 30$ , $x \approx 0.6066$ .