Paths in phase space

From the previous problem of the set, we can see that when we consider a bounded table and a ball placed at point $(0,0)$ to be rolled later and the vel. applied on the ball is $(0,1)$ m/s, then the movement is a periodic motion where the ball returns to the starting point $(0,0)$ after every $4$ seconds.

So, we can say that the ball returns to the original position at origin $O(0,0)$ after $\Large 4X$ seconds where $X$ is an arbitrary positive integer, i.e., $X=1,2,3,.....$ . So, if the time is a multiple of $4$ , then the ball is at the initial position again at $O$ .

Now, when we consider the no. of times ball bounces off the walls as $N$ and after $4N$ seconds, the ball is again at the initial position from where the ball was rolled off as $4N$ is a multiple of $4$ and this states that the ball is at original position within a circle of radius $L$ as the ball is actually not to start rolling from $O$ but in a region close to $O$ in a radius $L$ , here according to the problem and in this case, it can be observed that $\Large L=\epsilon$ . Since, it is itself said in the problem that the position is not accurately determined here, so we can say that the ball is at a circle of radius $\Large L=\epsilon$ of the origin after the total time $4N$ rather than our consideration of being at $O(0,0)$ as the ball is not starting to roll from $O$ in the first place. However, when a region is considered the ball will be in different positions after every $4$ sec, but since original starting location was near origin, so we can say that the final position tends to be at $O$ and more accurately say that the region is within a radius of $\large \epsilon$ from $O$ .

So, from all this we conclude that, $\Large \boxed{L=\epsilon}$