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).
   
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!

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).
   
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!

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).
   
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 $45^\circ$, 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!

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).
   
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!