Phase plane for a pendulum

This demonstration draws numerical approximations (using a Runge-Kutta method of order 4) to the phase curves for the system of equations \begin{align} \dot{x} = y \hspace{25pt}\\ \dot{y} = -\sin(x). \end{align} This system is equivlent to the single second-order equation \(\ddot{x} = -\sin(x)\), the equation for the motion of a pendulum.

Clicking on a any point in the frame will draw a phase curve beginning at that point. The "Continue phase curve" button will continue the current curve from where it ended; the "Reverse time" button will draw the phase curve for negative times, starting from the inital point of the last curve drawn.

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