# In a dynamical system, what might be a good reason why periodicity in an object's velocities is important?

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,

• A simple point is that if x is periodic, so too is xdot (with the same period) – dmuir Oct 12 '20 at 16:48

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 $$\tau$$.