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, where r is fixed. In what follows l is the constant length of the pendulum, so r = l.The Lagrangian is
The Euler–Lagrange equations give :
and
showing that angular momentum is conserved.
The conical pendulum refers to the special solutions where and is a constant not depending on time.The Hamiltonian is
where
and
