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.

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 )

Routinely, in order to write down the kinetic

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

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,

will play a role in the Hamiltonian formulation below.

The second order differential equation determining the evolution of

, 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

is a constant not depending on time.

The Hamiltonian is where conjugate momenta are and In terms of coordinates and momenta it reads

− m g l cos ⁡ θ

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

That is a consequence of the rotational symmetry of the system around the vertical axis.

[dubious – discuss] 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

lies between two circles of latitude,[1] where