Whirling pendulum

Nonlinearity makes the pendulum equation very difficult to solve analytically. The usual way around this is to fudge, by invoking the small angle approximation sin X X for x 1. This converts the problem to a linear one, which can then be solved easily. But by restricting to small x, we re throwing out some of the physics, like motions where the pendulum whirls over the top. Is it really necessary to make such drastic approximations ... 

Do you remember the first nonlinear system you ever studied in school It was probably the pendulum. But in elementary courses, the pendulum s essential nonlinearity is sidestepped by the small-angle approximation sin 6 d. Enough of that In this section we use phase plane methods to analyze the pendulum, even in the dreaded large-angle regime where the pendulum whirls over the top. 

There are several advantages to the cylindrical representation. Now the periodic whirling motions look periodic—they are the closed orbits that encircle the cylinder for E>1. Also, it becomes obvious that the saddle points in Figure 6.7.3 are all the same physical state (an inverted pendulum at rest). The heteroclinic trajectories of Figure... 

Now we re ready to tackle the full two-dimensional problem. As we claimed at the end of Section 4.6, for sufficiently weak damping the pendulum and the Josephson junction can exhibit intriguing hysteresis effects, thanks to the coexistence of a stable limit cycle and a stable fixed point. In physical terms, the pendulum can settle into either a rotating solution where it whirls over the top, or a stable rest state where gravity balances the applied torque. The final state depends on the initial conditions. Our goal now is to understand how this bistability comes about. 

Suppose we slowly decrease 7, starting from some value / > 1. What happens to the rotating solution Think about the pendulum as the driving torque is reduced, the pendulum struggles more and more to make it over the top. At some critical value 4 < 1, the torque is insufficient to overcome gravity and damping, and the pendulum can no longer whirl. Then the rotation disappears and all solutions damp out to the rest state. 

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