Verify that (0, 0), (, 0), and (2, 0) are all critical points of the first order

Question

Verify that (0, 0), (, 0), and (2, 0) are all critical points of the first order system (indeed, all (n, 0) will be critical points). Discuss the stability of the three given critical points by linearizing. Explain what each critical point means with respect to the position and velocity of the pendulum. Sketch a picture, be sure to include the three mentioned critical points and be sure to show a few trajectories,
especially those with larger initial velocities.

Need to see three linearizations (one about each of the critical points which is given), and then for each critical point, find the eigenvalues of the associated matrix, and make a conclusion about the stability of the system at that particular critical point.

Here's what I have so far:

Verify that (0, 0), (pi, 0), and (2pi, 0) are all critical points of the first order system (indeed, all (npi, 0) will be critical points). Discuss the stability of the three given critical points by linearizing. Explain what each critical point means with respect to the position and velocity of the pendulum. Sketch a picture, be sure to include the three mentioned critical points and be sure to show a few trajectories, especially those with larger initial velocities. Need to see three linearizations (one about each of the critical points which is given), and then for each critical point, find the eigenvalues of the associated matrix, and make a conclusion about the stability of the system at that particular critical point.

Explanation / Answer

The following link might help you. http://www-users.math.umd.edu/~rll/courses/math246.s11/246RS11-probsets2.pdf

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