Dynamics | Part IA, 2006

For the dynamical system

x¨=sinx\ddot{x}=-\sin x

find the stable and unstable fixed points and the equation determining the separatrix. Sketch the phase diagram. If the system starts on the separatrix at x=0x=0, write down an integral determining the time taken for the velocity x˙\dot{x} to reach zero. Show that the integral is infinite.

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