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The equations of motions for the double pendulum is given by

$$\dot{\theta_1} = \frac{6}{ml^2}\frac{2p_{\theta1} - 3\cos(\theta_1 - \theta_2)p_{\theta2}}{16 - 9\cos^2(\theta_1 - \theta_2)}$$

and similarly for the other pendulum. In respect to what does the change in angle for the first pendulum refer to? Is it with respect to time? So that $\dot{\theta_1} = \frac{d\theta}{dt}$?

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Yes. The point always refers to the derivative with respect to time.

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The dot over a function or variable Isaac Newton's notation for a derivative; in physics it always means a derivative with respect to time.

Variables with two or three dots, like $\ddot{\theta}$ and $\dddot{\theta}$, represent second and third time derivatives respectively.

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