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 solved using billiards...and I gave a really vague answer.

I'm looking for a precise example of a "real life" problem (besides the billiard game of course) that can be modeled using billiard dynamics.

"real life" = applied (i'm not english native speaker)

• According to Issac Asimov, killing your genius professional rival. – corsiKa Feb 6 '15 at 16:27
• It is very bad manners to ask a pure mathematician "what are the applications of your work"! :p – Federico Poloni Feb 6 '15 at 21:01
• @FerranV. --- en.wikipedia.org/wiki/The_Billiard_Ball – Carlo Beenakker Feb 7 '15 at 16:16
• It's a really bad idea to discourage and dismiss the few reporters who ask more about mathematics. Asking about the applications is not inherently rude. It is far better than avoiding mathematicians as much as possible when writing an article on mathematics and then inserting the statement that the mathematical progress involved lots of calculation even if the work's only numbers were page numbers, or making fun of how socially awkward the mathematicians seem, perpetuating misconceptions and stereotypes that make most reporters and readers more comfortable. – Douglas Zare Feb 9 '15 at 18:02
• @DouglasZare: I was being ironic...I think it is natural for everybody to ask for applications! This week I'm working with the reporter, the idea is to enrich together the final draft of the interview by adding some of the applications that others have gently posted here to answer my question (but explained in a more simple language). I think, at least in my country, that mathematics has a bad reputation among the general public, and this is a consequence (again, at least in my country) of the poor interaction of the mathematical community with the mass media. – Ferran V. Feb 9 '15 at 21:13

The billiard-ball computer, also known as a conservative logic circuit, is an idealized model of a reversible mechanical computer based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and Tommaso Toffoli. Instead of using electronic signals like a conventional computer, it relies on the motion of spherical billiard balls in a friction-free environment made of buffers against which the balls bounce perfectly. It was devised to investigate the relation between computation and reversible processes in physics.

The billiard-ball computer was never realized in this form, but it played a significant role in the development of the quantum computer. Since the unitary evolution of quantum mechanics is reversible, it cannot employ the irreversible logical operations of a conventional computer. (This story is told here.)

For an altogether different application of billiard ball dynamics, to semiconductor device physics, see Billiard model of a ballistic multiprobe conductor (1989).

• Thanks Carlo! This provides a good example, even though no device was ever realized in this form. – Ferran V. Feb 7 '15 at 15:22
• @CarloBeenakker links broken. – T.... Aug 12 at 14:08
• thanks for noting this --- broken link repaired. – Carlo Beenakker Aug 12 at 14:29

I am very surprised that nobody mentioned the REAL reason why mathematicians are so much interested in billiards. It comes from mechanics (celestial mechanics, first of all), and the first question considered was that of existence of periodic trajectories. Which correspond to periodic orbits in mechanics.

G. Birkhoff in "Dynamical systems" explains: "Before we give an example illustrating applications of Poincare theorem and its generalizations, let us consider first of all a special but very typical problem of this sort, namely the problem of motion of a billiard ball on a table which is bounded by a convex curve. This system is of a great interest for the following reasons. Every Lagrangian system with two degrees of freedom, ... etc."

For an easily available source, see Birkhoff, On the periodic motions of dynamical systems, Acta math., 48, 1927.

• Away from critical points, the flow of an ODE can be transformed to a field of parallel vectors. For larger regions this breaks down along a boundary. This means that one can transform a flow to go straight $\rightarrow$ "bounce" on boundary $\rightarrow$ go straight $\rightarrow$ another "bounce" $\rightarrow$ $\cdots$, which is what Birkhoff proves in the text of those ellipses. – Hbar Feb 25 '16 at 16:40

There are some potential applications to the design of optical cavities for lasers.

Imagine a region whose sides are mirrored. You might shine a laser in, and have an opening or a partial reflector where the light can come out. Typically, there is some "gain region" in the interior, say a crystal which is excited electrically or by a laser operating at another frequency. We want some input beam to pass through the gain region many times, or to spend a lot of time there on average, before the beam hits the output.

A complication not present in the usual mathematical billiards is that the gain region may have a different index of refraction from the surrounding medium, so light entering it at an angle may be deflected. This can even depend on the intensity of the light.

One possibility is to control the geometry very precisely. In fact, this is one place you can use the fact that hyperboloids of one sheet are doubly ruled surfaces: You can make the even segments follow one ruling while the odd segments follow the other ruling. However, aligning this precisely can be tricky, particularly with the complications above, and sometimes it doesn't make efficient use of the volume, so you don't get as many passes through the crystal for the space you allocate to the cavity.

You might want to design a cavity so that it tolerates small errors, so many paths spend a long time in the gain region and then exit.

Here is a surprising application of billiard-related mathematics. The story starts quite far, but rest assured that it will end up with problems related to billiard trajectories.

Consider the practical problem of finding out the position dependent electrical conductivity of an object by making voltage and current measurements at the boundary. This is known as Calderón's problem, and it can be formulated more precisely as follows. If the object is a domain $M\subset\mathbb R^3$, conductivity is modeled by a function $\gamma:M\to(0,\infty)$ that satisfies $\log\gamma\in L^\infty$ (measurable, bounded away from infinity and zero). If $u:M\to\mathbb R$ is the electric potential, the current density is by Ohm's law $-\gamma\nabla u$ and it follows from Kirchhoff's law that $\operatorname{div}(\gamma\nabla u)=0$. Measuring voltage at the boundary means measuring $u|_{\partial M}$ and measuring current amounts to measuring $\nu\cdot\gamma\nabla u|_{\partial M}$, where $\nu$ is the unit normal. The problem is then this: Given the voltage and current data $$\{(u|_{\partial M},\nu\cdot\gamma\nabla u|_{\partial M});u \in H^1(M),\operatorname{div}(\gamma\nabla u)=0\},$$ reconstruct the function $\gamma$.

If all of $\partial M$ is available for measurements, the problem is quite well understood — although much of the progress is very recent. In dimension two the problem can be solved (Astala–Päivärinta 2006) and also in higher dimensions with the additional assumption that $\gamma$ is Lipschitz (Caro–Rogers 2014).

In practical situations it usually happens that one cannot make perfect measurements. One case of imperfect data is a partial data problem where only part of $\partial M$ is available for measurements. On the rest of $\partial M$ the potential is assumed to vanish (grounded surface) and current cannot be measured. Suppose $M$ is contained in a cylinder $\mathbb R\times\Omega$ for a domain $\Omega\subset\mathbb R^2$ so that $M$ contains a cylindrical part $[0,L]\times\Omega$ and two caps at the ends to make it bounded. Assume the part inaccessible to measurements is contained in the cylindrical part, more precisely of the form $[0,L]\times R$ for $R\subset\partial\Omega$. It turns out that in this setting one can reconstruct a $C^2$ conductivity in $M$ if any function $f\in C(\Omega)$ can be recovered from its integrals over all billiard trajectories in $\Omega$ which reflect on $R$ and have endpoints on $\partial\Omega\setminus R$ (Kenig–Salo 2013).

This billiard problem (known as the broken ray tomography problem) is far from being completely understood, but the reconstruction is possible for example if $R$ is a subset of a cone (Ilmavirta 2014). Broken ray tomography provides open problems related to billiards and connected with practical applications in nondestructive imaging.

As well as the other excellent answers, try ray tracing, both in physics and engineering [propagation of waves (electrons, light, sound, ...)] and computer graphics [propagation of light for realistic image generation].

My survey article on the Lorentz gas (billiards in extended space), [http://arxiv.org/abs/1402.7010 and Commun. Theor. Phys.] gives a few more applications; the final paragraph reads

Physics of Transport Finally, the Lorentz gas and similar models have often been used to model transport on small scales. In this context, the use of polygonal channels for studying nanopores was mentioned in Subsec. 6.4. Lorentz channels have been used to understand thermoelectric efficiency.[220] Other examples have included confined fluids,[221−222] glasses,[7] nuclear collisions,[223] and zeolites.[224]

The keyword you are looking for is " Microorganism Billiards". Very recent topic, but now seems to be catching up in fluids/bio community.

Gregory Galperin invented billiard method of computing $\pi$, see Playing Pool With $\pi$ (The Number $\pi$ From A Billiard Point Of View)

To calculate $\pi$, take two identical balls. Put one near a wall and roll the other ball toward it. The first ball will hit the second, which will bounce off the wall and come back to hit the first ball. Click click click. Three collisions. The first digit of $\pi$ is a $3$!

A first ball that’s $100$ times as massive will create $31$ clicks. $10\,000$ times as massive will create $314$ clicks, etc.

• That is absolutely fantastic – Permian Apr 23 '18 at 18:49

The elastic collision mechanics of the cue ball and target billiard balls have an analog in the moderation of neutrons by hydrogen nuclei, in that they have nearly identical masses, so the physics of energy transfer are similar.

Point missed by everyone, is that the mathematics of billiards may well one day be called upon to save our species from extinction by an asteroid impact, and would therefore represent one of the greatest contributions of applied mathematics to mankind.

We may not be able to perturb the orbit of a large asteroid enough to avoid a collision in time, but we may be able to find one or more smaller ones and perturb their orbits enough to cannon off the large one with bigger effect.

Such an endeavour would require precise mathematics of momentum transfer and collision vectors taking into account relativity and the evaluation of thousands of candidates with trillions of potential solutions.

Not a trivial problem to solve, but I think necessary to have in the tool box.

• I don't think this is an application of billiard dynamics as understood in mathematics. I would not count all analysis of collisions as billiards. – Joonas Ilmavirta Feb 10 '15 at 15:26
• Or a big ball of garbage is plummeting to earth and we have to knock its trajectory with a ball of garbage of equal mass [/futurama]. – user78249 Nov 14 '16 at 19:37