  A 3D pendulum consists of a rigid body, supported at a fixed pivot, with
three rotational degrees of freedom. The pendulum is acted on by a
gravitational force. Symmetry assumptions are shown to lead to the planar 1D
pendulum and to the spherical 2D pendulum models as special cases. The case
where the rigid body is asymmetric and the center of mass is distinct from the
pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models
are introduced and used to study important features of the nonlinear dynamics:
conserved quantities, equilibria, invariant manifolds, local dynamics near
equilibria and invariant manifolds, and the presence of chaotic motions. These
results demonstrate the rich and complex dynamics of the 3D pendulum.
