For questions about billiards; dynamical systems involving point particles (billiard balls) that travel in straight lines on the interior of some bounded domain (the billiard table) and experience perfectly elastic reflections on collision with boundary.
Questions tagged [billiards]
69 questions
36
votes
2 answers
Reflection between Two Parallel Cylinders
I am interested in the following question.
Consider two identical cylinders (or in 2D, two circles) of radius $r$, with centers separated by a distance $s$. A point particle is released above with a vertical downward velocity, its initial horizontal…
Kuru Kurumi
- 411
- 2
- 6
33
votes
2 answers
Perfectly centered break of a perfectly aligned pool ball rack
This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed).
Imagine the beginning of a game of pool, you have 16…
Řídící
- 3,268
8
votes
2 answers
Bouncing Bullet Problem
This is a problem that was presented to me through the google foobar challenge, but my time has since expired and they had decided that I did not complete the problem. I suspect a possible bug on their side, because I passed 9/10 test cases (which…
Kraigolas
- 1,569
6
votes
1 answer
Proof for the existence of a second fixed point in Poincaré's last geometric theorem
In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem:
"An area-preserving transformation of an annulus that moves the boundary circles in opposite directions has at least two distinct fixed points."
He…
Klaas
- 709
6
votes
0 answers
Phase space and collision map of a billiard
I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions.
2.5. Phase space for the flow.
The state of a moving particle at any time is specified by its…
algae
- 198
6
votes
0 answers
Number of rays intersecting at a point inside a polygon
I'm working on a project to do with bouncing rays inside polygons and now I've reached a crucial stage of this project in which I need help with and is related to the problem stated below. Your help will be highly appreciated.
$\\$
Problem
Let $P_n$…
JCr
- 103
6
votes
2 answers
Launching billiard balls at 45 degree angles and bouncing of off edges.
Say I have a billiard ball and I launch it from the bottom-left corner of a table with length $x$ and width $y$. Given $x$ and $y$ will the ball reach a corner again, which corner and in how many times will it need to bounce off the edge.
*I launch…
mtheorylord
- 4,340
5
votes
3 answers
Polygonal billiards and uniform distribution
According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss…
brainjam
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5
votes
1 answer
If $D$ isn't a convex billiard table, then the billiard map $T$ is not continuous
I am currently reading chapter 3 of "Geometry and Billiards" by Serge Tabachnikov, and I have some doubts about the need of using convex sets.
So, here's how the billiard map is defined:
To fix ideas, consider a plane billiard table $D$ whose…
ImHackingXD
- 1,119
5
votes
0 answers
Why a measure over the points of a billiard which reflect infinitely often in finite time vanish, i.e. why $\int_{N\cap M}f(x^{-})d\mu_1(x^{-1})=0$
The crux of my question is that why $\mu(N^{(2)}) = 0$ when $N^{(2)}$ consists all points of a billiard that reflect infinitely often in finite time under a given flow (i.e. transformation) function. This is taken from the book Ergodic Theory by…
Epsilon Away
- 1,070
5
votes
1 answer
Do these balls collide?
Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when they hit the left/right wall, and similarly for the…
Martin Brandenburg
- 181,922
5
votes
1 answer
How to show that the Billiard flow is invariant with respect to the area form $\sin(\alpha)d\alpha\wedge dt$
Consider a plane billiard table $D \subset \mathbb{R}^2$ (i.e. a bounded open connected set) with smooth boundary $\gamma$ being a closed curve. Next, let $M$ denote the space of tangent unit vectors $(x,v)$ with $x$ on $\gamma$ and $v$ being a unit…
Quoka
- 3,131
5
votes
1 answer
Billiards in a holey square
Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because the point coordinates and direction vector are…
Joseph O'Rourke
- 31,079
4
votes
1 answer
For arbitrary billiard tables with elastic boundary reflections, is the "Lebesgue measure" an invariant of the flow maps?
This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) admit the Liouville measure (also known as the…
4
votes
1 answer
Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit
In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is:
How can I reproduce Schwartz's result using a simulation?
More specifically,
How can I…
rb3652
- 381