I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?

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    $\begingroup$ 2nd derivative with respect to the argument $\endgroup$
    – The Photon
    Oct 31, 2021 at 0:50
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    $\begingroup$ dot means derivative w/r to $t$, double-dot is two derivatives w/r to $t$. $\endgroup$ Oct 31, 2021 at 0:50
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    $\begingroup$ The dot for derivative with respect to time and double dot for second derivative with respect to time is standard notation for physics mechanics relationships. $\endgroup$
    – John Darby
    Oct 31, 2021 at 2:36
  • $\begingroup$ it changes the pronunciation of the angle, silly. $\endgroup$ Oct 31, 2021 at 4:16

1 Answer 1


It means the second time derivative.

In other words, $$\ddot\theta=\frac{d^2\theta}{dt^2}$$ which represents the angular acceleration of an object (which is a pendulum bob in your example).

These, and indeed first time derivatives (or even more than first, second etc.) are very common in physics (and in engineering and many other subjects), since we are often thinking about instantaneous timed rates of change of quantities. For example, the instantaneous rate in change of an objects position $x$ is called its instantaneous velocity $v$ where $$v=\dot x=\frac{dx}{dt}$$ and its acceleration is the rate in change of this quantity, or $$a=\ddot x=\frac{dv}{dt}=\frac{d^2x}{dt^2}$$


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