Math Insight

Applet: Undamped pendulum

The dynamical system describing an undamped pendulum requires two state variables. The angle $\theta$ of the pendulum and the angular velocity $\omega$. These two variables completely describe the state of the pendulum. The state space (or phase space) $(\theta,\omega)$ is illustrated in the left panel. The space is periodic in $\theta$, and the range $-\pi \le \theta < \pi$ is shown. The initial angle and angular velocity, $\vc{X}_0 = (\theta_0,\omega_0)=(\theta(0),\omega(0))$, is set by dragging the blue point. The resulting evolution of the system is illustrated both by the green curve (the trajectory) and the moving red point, which indicates the state (i.e., angle and angular velocity) at time $t$: $(\theta(t),\omega(t))$. Gray arrows represent the speed and direction that the point $(\theta(t),\omega(t))$ moves through the state space.

The right panel illustrates the evolution of just the angle $\theta(t)$ with respect to time $t$. The plot includes both the graph of $\theta$ versus $t$ in green and a moving red point, corresponding to the similar curve and point in the left panel. Above the graph are slider by which you can change the time $t$ and a graphical illustration of the pendulum with angle $\theta(t)$ and angular velocity $\omega(t)$.

If the angular velocity $\omega$ is sufficiently large, the pendulum will swing around past an angle of $\pm\pi$. In this case, the variable $\theta(t)$ will wrap around from $\pi$ to $-\pi$ or vice versa. At those points, the green curves will contain extraneous lines crossing between $\theta=\pi$ and $\theta=-\pi$. Those lines should be ignored.

Applet file: undamped_pendulum.ggb

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