Spherical pendulum

Spherical pendulum: angles and velocities.

In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.

Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed. In what follows l is the constant length of the pendulum, so r = l.

Lagrangian mechanics

Main article: Lagrangian mechanics

The Lagrangian is

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

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.

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

Hamiltonian mechanics

Main article: Hamiltonian mechanics

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

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

See also

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

References

• Weinstein, Alexander (1942). "The spherical pendulum and complex integration". The American Mathematical Monthly. 49 (8): 521–523. doi:10.1080/00029890.1942.11991275.
• Kohn, Walter (1946). "Countour integration in the theory of the spherical pendulum and the heavy symmetrical top". Transactions of the American Mathematical Society. 59 (1): 107–131. doi:10.2307/1990314. JSTOR 1990314.
• Olsson, M. G. (1981). "Spherical pendulum revisited". American Journal of Physics. 49 (6): 531–534. Bibcode:1981AmJPh..49..531O. doi:10.1119/1.12666.
• Horozov, Emil (1993). "On the isoenergetical non-degeneracy of the spherical pendulum". Physics Letters A. 173 (3): 279–283. Bibcode:1993PhLA..173..279H. doi:10.1016/0375-9601(93)90279-9.
• Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilizatio of its first integrals". Automatica. 40: 73–85. doi:10.1016/j.automatica.2003.07.009.
• 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.