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Periodic orbits of a billiard ball bouncing in a square have been well-studied. I am seeking similar analysis of what is sometimes called a rough ball, one whose high friction causes it to pick up spin when it hits a wall. Assuming no slip at the point of contact, that kinetic energy is conserved (a "superball"), and that gravity is not relevant, there is a definite dynamics, dependent upon the moment of inertia of the ball. For example, a solid ball (e.g., a lacrosse ball) has moment $I= \alpha m r^2$ where $\alpha=\frac{2}{5}$ (and $m$ and $r$ are mass and radius).
   SuperBall
As an example, if such a ball is thrown against the bottom side of a square, entering (along the red vector) with zero spin at $45^\circ$, horizontal velocity $1$, it exits at about $68^\circ$, with a clockwise spin resulting in a horizontal ball-rim velocity of $\frac{-10}{7}$. I've tried to track above the collision equations, without at all being certain that I am exactly correct. In my calculation, six bounces almost completes a cycle, but not quite.

Regardless of the accuracy of these calculations, my question is whether or not periodic orbits of such rough, elastic balls have been explored. Thanks!


(Added 10 Dec 2013.) Just found this paper by Aston & Shail: "The dynamics of a bouncing superball with spin." Dynamical Systems: An International Journal, Volume 22, Issue 3, 2007. Journal link.
   Fig9d
Blue trajectories have back-spin, red forward-spin. Obviously including dissipation. One cool aspect of this paper is that they analyze a ball bouncing up and down a staircase:
   Fig13
The dashed line is the initial "throw" that initiates the illustrated sequence of bounces.

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    $\begingroup$ This is not exactly what you need but, I believe, it's a good starting point: jdcowan.net/billiards/main_article.pdf $\endgroup$ Dec 8, 2013 at 3:14
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    $\begingroup$ Your diagram and equations for this dynamics appear to be given in fas.org/rlg/GarwinSuperBall.pdf . It is probably worth reading and summarising the papers that cite this. Another relevant field might be granular dynamics, but I expect most of these authors assume dissipation. $\endgroup$
    – user25199
    Dec 10, 2013 at 15:33
  • $\begingroup$ @Carl: Nice find! Their Fig6a is roughly the same as my figure, but they find a cycle, which indicates I may have made a computation error... $\endgroup$ Dec 10, 2013 at 17:34

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