a spherical pendulum

all of the pendula below are “planar” or have “hinge” joints that constrict motion to a plane. if we replace said hinge joints with a “ball-and-socket” joint, then we have what one calls a spherical pendulum. these have an increased range of motion and lead to some lovely behavior when viewing multiple initial conditions.

a simple spherical pendulum with multiple initial conditions

you might rightly complain that these “spherical” pendula do not take advantage of their increased degrees of freedom. if you look carefully, each pendulum above moves within a single plane. this is a consequence of there being no initial velocity.

let’s change that, and see what motion ensues…

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