Spherical pendulum

A spherical pendulum is a generalization of the pendulum. It consists of a mass moving without friction on a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.

It is convenient to use spherical coordinates and describe the position of the mass in terms of $(r,\theta,\phi)$ , where r is fixed.

L=\frac{1}{2} mr^2\left( \dot{\theta}^2+\sin^2\theta\ \dot{\phi}^2 \right) + mgr\cos\theta.

The Euler–Lagrange equations give :

\frac{d}{dt} \left(mr^2\dot{\theta} \right) -mr^2\sin\theta\cos\theta\dot{\phi}^2+ mgr\sin\theta =0

and

\frac{d}{dt} \left( mr^2\sin^2\theta \, \dot{\phi} \right) =0

showing that angular momentum is conserved.

And the Hamiltonian is

H=P_\theta\dot \theta + P_\phi\dot \phi-L

where

P_\theta=\frac{\partial L}{\partial \dot \theta}=mr^2\dot \theta

and

P_\phi=\frac{\partial L}{\partial \dot \phi} = mr^2\dot \phi \sin^2 \theta

References

↑ Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960.