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.