Spherical pendulum

The Lagrangian is [1]

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

The Euler–Lagrange equations give :

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

and

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

showing that angular momentum is conserved.

The conical pendulum refers to the special solutions where ${\displaystyle {\dot {\theta }}=0}$ and ${\displaystyle {\dot {\phi }}}$ is a constant not depending on time.

The Hamiltonian is

${\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L}$

where

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

and

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

• Conical Pendulum
• Newton's three laws of motion
• Pendulum
• Pendulum (mathematics)
• Routhian mechanics

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• Olsson, M. G. (1981). "Spherical pendulum revisited". American Journal of Physics. 49 (6): 531–534. Bibcode:1981AmJPh..49..531O. doi:10.1119/1.12666.
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• Essen, Hanno; Apazidis, Nicholas (2009). "Turning points of the spherical pendulum and the golden ration". European Journal of Physics. 30 (2): 427–432. Bibcode:2009EJPh...30..427E. doi:10.1088/0143-0807/30/2/021.
• Dullin, Holger R. (2013). "Semi-global symplectic invariants of the spherical pendulum". Journal of Differential Equations. 254 (7): 2942–2963. doi:10.1016/j.jde.2013.01.018.