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I'm studying periodic motions in a dynamical system and, as a newbie, I narrowly think of an object's periodicity in its spatial x-y coordinates, but what might be a good reason why the existence of periodicity in its velocities xdot and ydot is important too? I tend to think that so long as the object's spatial x and y coordinates repeat, we have nice time-periodic behavior, but I don't normally think of the object's velocities repeating as anything significant or sought after.

Any references (book / papers) are welcome too.

Thanks,

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  • $\begingroup$ A simple point is that if x is periodic, so too is xdot (with the same period) $\endgroup$ – dmuir Oct 12 '20 at 16:48
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Think of an object that moves along a spiral -- say, an electron moving in a uniform magnetic field. Its positions are not periodic (it never comes back to its original place) but its velocities are. Looking at periodicity in the velocities therefore tells us something we don't know by just looking at the positions.

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If a certain velocity component is periodic with period $\tau$ that means that the corresponding coordinate, as a function of time, is a sum of a linear function and a periodic function with the same period.

To prove it, let $\dot{x}(t)$ be periodic, $\dot{x}(t+\tau)=\dot{x}(t)$, and the integral over the period $\int_{t}^{t+\tau} \dot{x} dt = I$, where $I$ is independent of $t$.

Next, consider $y(t) = x(t) - I t/\tau$. Since $\dot{y} = \dot{x} - I/\tau$ it follows that $\dot{y}$ is also periodic with period $\tau$. Also, $\int_{t}^{t+\tau} \dot{y} dt = \int_{t}^{t+\tau} \dot{x} dt - I = 0$. Therefore, $y(t+\tau)-y(t)=0$, so $y(t)$ is periodic with period $\tau$.

Thus we arrive at the conclusion that $x(t)$ is a sum of a linear function $I t /\tau$ and a periodic function with period $\tau$.

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