The top of a pendulum of length

hangs from the origin. The mass

at the bottom end of the pendulum has coordinates

,

,

, where the vector

from the origin to

is at an angle θ to the negative

axis. The spherical coordinates of

are (

,

,

) with

. The Lagrange function and equations give

,

, and

. The integration constants are

,

,

, and the angular momentum

. The movement of the spherical pendulum is constrained to the spherical shell between

and

for all

values. The pendulum cannot reach the singular points

and

for

. When the angular momentum vanishes, the pendulum moves in a plane.