All Questions
Tagged with billiards ds.dynamical-systems
45 questions
1
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96
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Periodic orbits in planar smooth billiard table with large periods
Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering ...
- 441
2
votes
0
answers
109
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Proving light escapes mirrors via ergodic theory of billiards
There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
- 509
3
votes
0
answers
145
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2-ball billiards in a circle
Consider a 2D circular billiards table with diameter 1m containing two
balls with diameter 0.25m. Let each ball start with a speed of 1m/s.
In general, this speed could change after the balls hit ...
- 1,547
5
votes
2
answers
277
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Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...
- 649
5
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0
answers
166
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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
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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
33
votes
3
answers
2k
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Why is the billiard problem for obtuse triangles so hard?
This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open ...
- 1,087
1
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0
answers
84
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Trapping lightrays under nonstandard reflections and/or paths
Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
- 151k
4
votes
1
answer
616
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Types of triangles admitting periodic billiard orbits
It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and obtuse ...
- 1,087
2
votes
1
answer
139
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6-periodic billiards trajectory in acute triangle
We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?
- 33
8
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1
answer
322
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Under which conditions do ellipsoids have a focal property?
Given an ellipsoid $E$, we consider the trajectories of light inside $E$ assuming that $\partial E$ would be a mirror. In other words, let a light trajectory be piecewise linear path $\gamma:[0,\infty)...
18
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480
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Trapping lightrays with segment mirrors
Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors?
I posed this question in several forums before (e.g., here
and in an ...
- 151k
3
votes
2
answers
194
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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
39
votes
2
answers
3k
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3D Billiards problem inside a torus
I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...
- 589
10
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1
answer
324
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A question about billiards
This is a question in a rather well investigated subject of which I know very little and I have a hard time "translating" the general results available. Let me also say that I got interested in this ...
- 34.7k
9
votes
3
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612
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Existence of periodic orbits in rational billiards
Recently I've got interested in dynamical billiards. Some results in this field are obtained by elementary methods. For instance, see George W. Tokarsky's Polygonal Rooms Not Illuminable from Every ...
- 315
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1
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579
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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
1
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0
answers
38
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Boundedness of partial products for a divergent trig product
I am looking at a discrete dynamical system and I wish to show that it is bounded. I know that the displacement after $n$ iterations is given by the product
$$\Delta_n=\prod_{k=0}^n \left(1+\frac{2\...
- 11
6
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0
answers
180
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How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?
Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the ...
5
votes
2
answers
319
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Are periodic billiard trajectories stable on a manifold with strictly convex boundary?
Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary).
...
- 8,086
3
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0
answers
171
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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
10
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3
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1k
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Which polygons have *simple* periodic billiard paths?
I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Howard Masur proved in the 1980's that every rational polygon
(vertex angles rational ...
- 151k
37
votes
11
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7k
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What "real life" problems can be solved using billiards?
Recently I gave an interview to local media where I explained some basic open problems in billiard dynamics.
After a 45 min interview the reported asked me what "real life" problems can be ...
- 637
5
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1
answer
189
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Unfoldings of trajectories on the Veech triangle $V_4$
Let $V_4$ be the isosceles triangle with base angle $\pi/8$. $V_4$ is a Veech triangle, so the dynamics of billiards on it are very well understood.
Above is the unfolding of $V_4$, with edge ...
- 881
8
votes
4
answers
666
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Existence of nonergodic polygonal billiard
Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.
A standard conjecture is that a ...
- 2,696
8
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1
answer
694
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Raphael Douady's thesis: Applications du théorème des tores invariants
Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts.
According Wikipedia, he proves of the equivalence of KAM ...
- 2,244
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
7
votes
2
answers
274
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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
1
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1
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270
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Computing saddle connections in flat structures
Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a ...
- 881
32
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5
answers
1k
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Can every $\mathbb{Z}^2$ disk be pinball-reached?
Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be hit ...
- 151k
4
votes
0
answers
83
views
Do identical orbit tiles imply identical combinatorial types?
Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...
- 881
10
votes
1
answer
447
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Does the $n$-gonal billiards conjecture follow from the $m$-gonal conjecture when $m>n$?
For $n\ge 3$, define the $n$-gonal billiards conjecture as the statement
All convex $n$-gons admit periodic billiard trajectories.
To the best of my knowledge this question remains open for all $n$...
- 881
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
31
votes
3
answers
3k
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Optic fibers after Joseph O'Rourke
Let $\gamma\colon[a,b]\to \mathbb R^3$ be a smooth curve with curvature $< 1$.
Consider a tube, formed by one parameter family of unit circles with center at $\gamma(t)$ in the plane orthogonal to $...
- 45k
3
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0
answers
262
views
Polygon illumination with perturbed reflections
Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then ...
- 85.6k
37
votes
6
answers
3k
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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
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
7
votes
2
answers
593
views
Does the random Lorenz gas have a non-trivial diffusion coefficient?
For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not,...
- 599
24
votes
2
answers
1k
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Billiard dynamics for multiple balls
I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
...
- 151k
8
votes
0
answers
246
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Billiards with incompatible regions
An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong counter-examples:...
- 131
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3
answers
2k
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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
33
votes
4
answers
3k
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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
6
votes
0
answers
450
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Differential equation of line tangent to caustics
This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...
- 281
14
votes
2
answers
1k
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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
9
votes
4
answers
2k
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Birkhoff conjecture about integrable billiards
There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse.
Integrability here might be ...
- 8,425