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.
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