Digital Library of Mathematical Functions
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22 Jacobian Elliptic FunctionsApplications22.19 Physical Applications

Figure 22.19.1 (See in context.)

See accompanying text
Figure 22.19.1: Jacobi’s amplitude function \mathop{\mathrm{am}\/}\nolimits\left(x,k\right) for 0\leq x\leq 10\pi and k=0.5,0.9999,1.0001,2. When k<1, \mathop{\mathrm{am}\/}\nolimits\left(x,k\right) increases monotonically indicating that the motion of the pendulum is unbounded in \theta, corresponding to free rotation about the fulcrum; compare Figure 22.16.1. As k\to 1-, plateaus are seen as the motion approaches the separatrix where \theta=n\pi, n=\pm 1,\pm 2,..., at which points the motion is time independent for k=1. This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. For k>1, the motion is periodic in x, corresponding to bounded oscillatory motion.
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