The equation of motion of an undamped pendulum is d2/dt2 +

Chapter 9, Problem 21

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The equation of motion of an undamped pendulum is d2/dt2 + 2 sin = 0, where2 = g/L. Let x = , y = d/dt to obtain the system of equationsdx/dt = y, dy/dt = 2 sin x.(a) Show that the critical points are (n, 0), n = 0, 1, 2, ... , and that the system is locallylinear in the neighborhood of each critical point.(b) Show that the critical point (0, 0) is a (stable) center of the corresponding linearsystem. Using Theorem 9.3.2, what can you say about the nonlinear system? Thesituation is similar at the critical points (2n, 0), n = 1, 2, 3, .... What is the physicalinterpretation of these critical points?(c) Show that the critical point (, 0) is an (unstable) saddle point of the correspondinglinear system. What conclusion can you draw about the nonlinear system? Thesituation is similar at the critical points[(2n 1), 0], n = 1, 2, 3, ....What is the physicalinterpretation of these critical points?(d) Choose a value for 2 and plot a few trajectories of the nonlinear system in theneighborhood of the origin. Can you now draw any further conclusion about the natureof the critical point at (0, 0) for the nonlinear system?(e) Using the value of 2 from part (d), draw a phase portrait for the pendulum. Compareyour plot with Figure 9.3.5 for the damped pendulum.