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

Figure 22.19.1 (See in context.)

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Figure 22.19.1: Jacobi’s amplitude function am(x,k) for 0x10π and k=0.5,0.9999,1.0001,2. When k<1, am(x,k) increases monotonically indicating that the motion of the pendulum is unbounded in θ, corresponding to free rotation about the fulcrum; compare Figure 22.16.1. As k1-, plateaus are seen as the motion approaches the separatrix where θ=nπ, n=±1,±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.