Spherical pendulum

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.

Spherical pendulum: angles and velocities.

Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of $(r,\theta ,\phi )$, where r is fixed such that $r=l$.

Lagrangian mechanics

Main article: Lagrangian mechanics

Routinely, in order to write down the kinetic $T={\tfrac {1}{2}}mv^{2}$ and potential $V$ parts of the Lagrangian $L=T-V$ in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here, following the conventions shown in the diagram,

$x=l\sin \theta \cos \phi$

$y=l\sin \theta \sin \phi$

$z=l(1-\cos \theta )$.

Next, time derivatives of these coordinates are taken, to obtain velocities along the axes

${\dot {x}}=l\cos \theta \cos \phi \,{\dot {\theta }}-l\sin \theta \sin \phi \,{\dot {\phi }}$

${\dot {y}}=l\cos \theta \sin \phi \,{\dot {\theta }}+l\sin \theta \cos \phi \,{\dot {\phi }}$

${\dot {z}}=l\sin \theta \,{\dot {\theta }}$.

Thus,

$v^{2}={\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}=l^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\phi }}^{2}\right)$

and

$T={\tfrac {1}{2}}mv^{2}={\tfrac {1}{2}}ml^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\phi }}^{2}\right)$

$V=mg\,z=mg\,l(1-\cos \theta )$

The Lagrangian, with constant parts removed, is

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

The Euler–Lagrange equation involving the polar angle $\theta$

${\frac {d}{dt}}{\frac {\partial }{\partial {\dot {\theta }}}}L-{\frac {\partial }{\partial \theta }}L=0$

gives

${\frac {d}{dt}}\left(ml^{2}{\dot {\theta }}\right)-ml^{2}\sin \theta \cdot \cos \theta \,{\dot {\phi }}^{2}+mgl\sin \theta =0$

and

${\ddot {\theta }}=\sin \theta \cos \theta {\dot {\phi }}^{2}-{\frac {g}{l}}\sin \theta$

When ${\dot {\phi }}=0$ the equation reduces to the differential equation for the motion of a simple gravity pendulum.

Similarly, the Euler–Lagrange equation involving the azimuth $\phi$,

${\frac {d}{dt}}{\frac {\partial }{\partial {\dot {\phi }}}}L-{\frac {\partial }{\partial \phi }}L=0$

gives

${\frac {d}{dt}}\left(ml^{2}\sin ^{2}\theta \cdot {\dot {\phi }}\right)=0$.

The last equation shows that angular momentum around the vertical axis, $|\mathbf {L} _{z}|=l\sin \theta \times ml\sin \theta \,{\dot {\phi }}$ is conserved. The factor $ml^{2}\sin ^{2}\theta$ will play a role in the Hamiltonian formulation below.

The second order differential equation determining the evolution of $\phi$ is thus

${\ddot {\phi }}\,\sin \theta =-2\,{\dot {\theta }}\,{\dot {\phi }}\,\cos \theta$.

The azimuth $\phi$, being absent from the Lagrangian, is a cyclic coordinate, which implies that its conjugate momentum is a constant of motion.

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 conjugate momenta are

$P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}$

and

$P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}$.

In terms of coordinates and momenta it reads

$$H=\underbrace {\left[{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}\right]} _{T}+\underbrace {{\bigg [}-mgl\cos \theta {\bigg ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta$$

Hamilton's equations will give time evolution of coordinates and momenta in four first-order differential equations

${\dot {\theta }}={P_{\theta } \over ml^{2}}$

${\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }$

${\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta$

${\dot {P_{\phi }}}=0$

Momentum $P_{\phi }$ is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.

Trajectory

Trajectory of a spherical pendulum.

Trajectory of the mass on the sphere can be obtained from the expression for the total energy

$E=\underbrace {\left[{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}\right]} _{T}+\underbrace {{\bigg [}-mgl\cos \theta {\bigg ]}} _{V}$

by noting that the horizontal component of angular momentum $L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}$ is a constant of motion, independent of time. This is true because neither gravity nor the reaction from the sphere act in directions that would affect this component of angular momentum.

Hence

$E={\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta$

$\left({\frac {d\theta }{dt}}\right)^{2}={\frac {2}{ml^{2}}}\left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]$

which leads to an elliptic integral of the first kind for $\theta$

$t(\theta )={\sqrt {{\tfrac {1}{2}}ml^{2}}}\int \left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]^{-{\frac {1}{2}}}\,d\theta$

and an elliptic integral of the third kind for $\phi$

$\phi (\theta )={\frac {L_{z}}{l{\sqrt {2m}}}}\int \sin ^{-2}\theta \left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]^{-{\frac {1}{2}}}\,d\theta$.

The angle $\theta$ lies between two circles of latitude, where

$E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta$.

See also

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

Further reading

• 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). "Contour 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.
• Richter, Peter H.; Dullin, Holger R.; Waalkens, Holger; Wiersig, Jan (1996). "Spherical pendulum, actions and spin". J. Phys. Chem. 100 (49): 19124–19135. doi:10.1021/jp9617128. S2CID 18023607.
• Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilization 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 ratio". European Journal of Physics. 30 (2): 427–432. Bibcode:2009EJPh...30..427E. doi:10.1088/0143-0807/30/2/021. S2CID 121216295.
• Dullin, Holger R. (2013). "Semi-global symplectic invariants of the spherical pendulum". Journal of Differential Equations. 254 (7): 2942–2963. arXiv:1108.4962. Bibcode:2013JDE...254.2942D. doi:10.1016/j.jde.2013.01.018.