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What is the general mathematical definition of periodic motion?

My answer : Can you say that each function is in the form $ \overrightarrow{r} (t)= \overrightarrow{r} (t+T)$.

it is right?

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https://www.britannica.com/science/periodic-motion

"Periodic motion, in physics, motion repeated in equal intervals of time. Periodic motion is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave. ... Simple harmonic motion is a special case of periodic motion."

So to answer your question $\overrightarrow{r} (t)= \overrightarrow{r} (t+T)$ describes periodic motion.

See https://brilliant.org/wiki/identifying-periodic-motion/ which has your exact equation as the mathematical definition of periodic motion.

{However damped harmonic motion would not be periodic motion since it does not exactly repeat because of the damping factor}

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