All Questions
Tagged with ds.dynamical-systems billiards
13 questions
100
votes
6
answers
5k
views
Light rays bouncing in twisted tubes
Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...
- 151k
96
votes
2
answers
114k
views
Perfectly centered break of a perfectly aligned pool ball rack
Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...
- 999
37
votes
6
answers
3k
views
Billiard dynamics under gravity
Has the dynamics of billiards in a polygon subject to gravity been
studied?
What I have in mind is something like this:
Still Snell's Law ...
- 151k
33
votes
4
answers
3k
views
Does there exist a shot in ideal pocket billiards?
Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...
- 151k
25
votes
4
answers
1k
views
Pinball on the infinite plane
Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts just touching the origin pin, and shoots off at angle $\...
- 151k
14
votes
2
answers
1k
views
Polygonal billards programs
I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
(source)
It was a good exercise, but at this point I ...
- 22.8k
10
votes
1
answer
579
views
Periodic billiard paths in hyperbolic triangles
It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path,
one obeying the reflection law that the angle of incidence equals the angle of reflection. ...
- 151k
7
votes
2
answers
274
views
Well-definedness of single-particle smooth billiards flow
Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well known;...
- 19.7k
5
votes
3
answers
2k
views
Dense orbits in billiards
This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ...
- 445
5
votes
0
answers
166
views
Pocket billiards with balls in general position
There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does ...
- 151k
4
votes
0
answers
232
views
Illuminating a just-barely irrational polygon
As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Let $P$ be a rational polygon.
Then for ...
- 151k
3
votes
0
answers
171
views
Trapped Billiard trajectories on non-convex billiard tables
Let $\Omega$ be a domain in $\mathbb{R}^2$ with smooth boundary. A billiard trajectory is a continuous curve $c: \mathbb{R}\supseteq I \longrightarrow \overline{\Omega}$ such that
$c(t) \in \partial ...
- 9,853
3
votes
2
answers
194
views
A Really Simple Stochastic Dynamic Billiard
Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, $c>0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $t$ of a point which moves ...
- 640