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1: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

►With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is …The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta \in (0,\pm \alpha )$ and $\theta \in (\pm \alpha ,0)$ have the same “quarter periods” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►

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Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. When , $\mathrm{am}(x,k)$ 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. …This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. … Magnify

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2: 28.33 Physical Applications

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§28.33(iii) Stability and Initial-Value Problems

►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\cos \left(2\omega t\right)$. …