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 , where r is fixed such that .

Lagrangian mechanics

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Routinely, in order to write down the kinetic   and potential   parts of the Lagrangian   in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here, following the conventions shown in the diagram,

 
 
 .

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

 
 
 .

Thus,

 

and

 
 

The Lagrangian, with constant parts removed, is[1]

 

The Euler–Lagrange equation involving the polar angle  

 

gives

 

and

 

When   the equation reduces to the differential equation for the motion of a simple gravity pendulum.

Similarly, the Euler–Lagrange equation involving the azimuth  ,

 

gives

 .

The last equation shows that angular momentum around the vertical axis,   is conserved. The factor   will play a role in the Hamiltonian formulation below.

The second order differential equation determining the evolution of   is thus

 .

The azimuth  , 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   and   is a constant not depending on time.

Hamiltonian mechanics

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The Hamiltonian is

 

where conjugate momenta are

 

and

 .

In terms of coordinates and momenta it reads

 

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

 
 
 
 

Momentum   is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.[dubiousdiscuss]

Trajectory

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Trajectory of a spherical pendulum.

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

 

by noting that the horizontal component of angular momentum   is a constant of motion, independent of time.[1] This is true because neither gravity nor the reaction from the sphere act in directions that would affect this component of angular momentum.

Hence

 
 

which leads to an elliptic integral of the first kind[1] for  

 

and an elliptic integral of the third kind for  

 .

The angle   lies between two circles of latitude,[1] where

 .

See also

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References

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

Further reading

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