Ian Jauslin

## Pendulum

$$U(\theta)=-\omega^2\cos(\theta),\qquad \omega=2$$ The graphs below show the phase space, potential, and a physical representation of the pendulum in motion. The energy can be adjusted with the slider below and exhibit the different types of motion:

• for $E\in(-\omega^2,\omega^2)$: oscillations
• for $E\in(\omega^2,\infty)$: rotation
• for $E=-\omega^2$: constant motion
• for $E=\omega^2$: separatrix: the system requires an infinite amount of time to reach the unstable equilibrium points $\pm\pi$. Because of numerical errors in the simulation the time is not infinite, but large.

Energy: E=-0.5

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