Friday, 3 November 2017

classical mechanics - How can a pendulum have amplitude greater than $pi$?


How can a pendulum have amplitude angle greater than $\pi$? I've been reading about phase plots, which are graphs of the $\frac{d\theta}{dt}$ on the $y$ axis and $\theta$ on the $x$ axis, shown below.


enter image description here


I can understand that the curves are not perfect ovals because we cannot use the small angle approximation. I can also see that there is one curve in the right side's drawing which looks like a sine curve shape, but intersects the $x$ axis at $\pi$ and $-\pi$.


But how are the other curves i.e. the curve intersecting $y$ axis at +2 created? What equation gives that and how would the pendulum's motion look? How would such an equation be derived?


I have a pendulum equation $$\frac{d\theta}{dt}=\pm\sqrt{\frac{2g}{l}(1-cos\theta_A)}$$ ($\theta_A$ is amplitude)


I derived that with conservation of energy laws, like for a simple pendulum but I did not do the small angle approximation. I tried to put $\theta_A=\pi+1$ and I got a simple oval, not the pair of curves that is symmetric about the $x$ axis. I do not understand the math and notation being used in this question: What is the period of a physical pendulum without using small-angle approximation?


I'm confused.




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