Figure 22.19.1 (See in context.)Figure 22.19.1: Jacobi’s amplitude function
for
and .
When , increases monotonically indicating
that the motion of the pendulum is unbounded in ,
corresponding to free rotation about the fulcrum; compare Figure
22.16.1. As , plateaus are seen as the motion
approaches the separatrix where ,
, at which points the motion is time
independent for . This corresponds to the pendulum being
“upside down” at a point of unstable equilibrium.
For , the motion is periodic in , corresponding to bounded
oscillatory motion.
