Wikipedia / Spherical pendulum
Wikipedia / 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.
It is convenient to use the spherical coordinates to describe the position of the mass in terms of , where .
Lagrangian mechanics
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
gives
and
the equation reduces to the differential equation for the motion of a simple gravity pendulum.
,
gives
.
The last equation shows that angular momentum will play a role in the Hamiltonian formulation below.
is thus
.
, being absent from the Lagrangian, is a cyclic coordinate, which implies that its conjugate momentum is a constant of motion.
The conical pendulum is a constant not depending on time.
Hamiltonian mechanics
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
is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.
Trajectory
Trajectory of the mass on the sphere can be obtained from the expression for the total energy
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
.
lies between two circles of latitude, where
.