Spherical pendulum

In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass 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, where is fixed such that.

Lagrangian mechanics

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
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.

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.

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. 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 for
and an elliptic integral of the third kind for
The angle lies between two circles of latitude, where