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Let us fix a reference frame $S$ with origin in $O$ in the euclidean space $\Bbb R^3$, then let us also define a periodic motion in the following manner:

A motion is periodic if and only if the time-dependant position with respect to $O$, $\underline{r}(t)$ is a periodic function, that is: $$ \exists T \in \Bbb R: \underline{r}(t+T) = \underline{r}(t), \forall t \in \Bbb R$$

Now, I have read somewhere on my textbook the following conditions:

  • Necessary Condition: the trajectory must be a closed curve;
  • Sufficient Condition: a closed-trajectory motion is periodic if it is uniform (the scalar velocity is constant).

My question is: are such conditions true, and if so, why?

I have tried answering them on my own but didn't succeed in getting a proof nor a counter-example. As always, let me know if I can explain myself more clearly and thank you for any answer or comment.

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    $\begingroup$ You mean whether or not the conditions are true for periodic motion? $\endgroup$
    – schris38
    Commented Jun 25, 2022 at 20:36
  • $\begingroup$ @schris38 I mean whether the necessary condition is true assuming the motion is periodic and whether the sufficient condition is enough to prove a motion to be periodic $\endgroup$
    – Matteo Menghini
    Commented Jun 26, 2022 at 10:18
  • $\begingroup$ It seems that this definition of periodic motion includes objects traveling in a linear open path at constant velocity since a constant velocity is a periodic translation. Is this what you intended? $\endgroup$
    – Stevan V. Saban
    Commented Jun 26, 2022 at 16:56

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