The caustics only an ellipse keeps

Equal diagonals in a curved quadrilateral ended with a remark about billiards. The confocal conics — the ellipses and hyperbolas that share two foci, which two families that cross at right angles laid out as a coordinate grid — are the curves a billiard ball in an elliptical table keeps touching. Every straight segment of a ball’s path is tangent to one of them, the same one for the whole of the ball’s life. A table with that property is called integrable, and the ellipse and the circle, an ellipse whose foci coincide, are the only tables known to have it. George Birkhoff asked in the 1920s whether they are the only ones there are.

This essay makes that question concrete by running the billiard. It confirms the ellipse’s property to the last digit a computer carries, then bends the table by a percent and a half and watches what the bend destroys. Most of the order survives, deformed but intact. One family of paths breaks into islands, a band near the foci dissolves into chaos, and the size of the damage follows a clean law in the size of the bend. That is the picture behind Birkhoff’s question, and behind the reason it has resisted proof for a century: a bend can be made small enough that nearly all the order survives, and still not all of it does.

The curve every segment touches #

The table is the ellipse $x^2 + y^2/B^2 = 1$ with $B = 0.7$, so its foci sit at $\pm\sqrt{1 - B^2}$, about $\pm 0.714$. A ball leaves a point on the wall in a straight line, meets the wall again, and reflects so that the angle of incidence equals the angle of reflection — the rule that every ray comes back to the other focus used to send a ray from one focus through the other.

The hero figure shows the two kinds of path that result. A path that never crosses the segment between the foci winds round the table, and all of its segments are tangent to a smaller ellipse with the same foci; the region inside that ellipse is never entered. A path that crosses the segment between the foci does so on every segment, and all of its segments are tangent to the two branches of a hyperbola with the same foci; the regions beyond the branches, near the ends of the long axis, are never entered. Paths that pass through a focus pass alternately through both, and form the boundary between the two kinds. In each case the curve the segments touch is called the path’s caustic, after the bright curves light makes when it is concentrated by a curved mirror.

The caustic can be named by one number. A confocal conic has the form $x^2/(1 - \lambda) + y^2/(B^2 - \lambda) = 1$, an ellipse when $\lambda < B^2$ and a hyperbola when $B^2 < \lambda < 1$. A line through a point $P$ with unit direction $d$ is tangent to the conic with parameter $\lambda$ exactly when

$$\lambda = d_y^2 + B^2 d_x^2 - (P \times d)^2,$$

where $P \times d$ is the cross product $P_x d_y - P_y d_x$. Billiards in an ellipse conserves this quantity: whatever $\lambda$ the first segment has, every later segment has too. The classical proof uses the reflection property of the foci and the fact that the tangent lines from a point to two confocal conics make equal angles with the lines to the foci. In a simulation it can simply be watched.

Smooth curves everywhere #

The way to see a billiard all at once is to plot each bounce as a point in a plane. Across is where on the wall the bounce happens, measured by the polar angle of the bounce point; up is the cosine of the angle the outgoing path makes with the wall, which is near one for a path that hugs the wall and near nought for one that leaves at right angles. Birkhoff introduced these coordinates, and in them the bounce map preserves area — the same property that made the standard map and every twist map behave as they do.

In the ellipse every path’s bounces fall on a single smooth curve, because a conserved quantity confines them there: the curve is the set of points with the same $\lambda$. The curves come in two families, separated by a figure-of-eight through the paths that pass through the foci. Across the top and bottom of the picture are the paths that wind round the table, with ellipse caustics; the ones nearest the top hug the wall, like a whisper travelling round a gallery. In the two lobes are the paths with hyperbola caustics, which oscillate across the short axis in closed loops round its centre. The centre of each lobe is the two-bounce orbit that runs up and down the short axis, which is stable: a path that starts near it stays near it. The crossing points of the figure-of-eight are the two-bounce orbit along the long axis, which is unstable, and the figure-of-eight itself is made of paths through the foci, which approach that unstable orbit for ever without reaching it.

There is no other kind of motion. The picture is filled, everywhere, with smooth curves, and there is no region in which a path can wander. That is what integrability means in practice: the whole space of motions is foliated by curves along which each motion is confined.

The quantity, watched #

The conservation can be checked directly. The next figure follows $\lambda$ through 150 bounces of one path, which leaves the top of the table with its direction making a tangential component of $0.9$ with the wall.

In the ellipse, $\lambda = 0.19$ and it stays there. Its spread over the run is about $3 \times 10^{-15}$, which is the rounding error of double-precision arithmetic: the confocal ellipse with that parameter is touched by every segment to every digit the machine keeps.

Now bend the table. The bent tables used from here on are $x^2 + y^2/B^2 + \varepsilon x^4 = 1$, with a small number $\varepsilon$ setting the bend. For $\varepsilon = 0.03$ the table’s long half-axis shrinks from $1$ to about $0.985$, a change of a percent and a half, and the short half-axis does not change; the table is still smooth, convex and symmetric in both axes, and drawn beside the ellipse the two are hard to tell apart. The same quantity, computed after each bounce in the bent tables, no longer stays put. With $\varepsilon = 0.005$ it oscillates over a range of $0.006$, and with $\varepsilon = 0.03$ over $0.033$. The bent table has no confocal family for $\lambda$ to name. Whether it has some other conserved quantity, some other family of curves that its paths keep touching, is the real question, and the next figure answers it for most paths and not for all.

Islands where a curve was #

The phase portrait of the table bent by $\varepsilon = 0.03$ is drawn with the same coordinates and the same starting paths, plus a few more aimed at the places where something has changed.

Most of the picture looks as it did. The curves near the top and bottom, the whispering-gallery paths, are still there, slightly wavy. The loops in the two lobes are still there. But two things are new, and both are what the theory of nearly integrable systems says must happen.

The first is a chain of islands. In the ellipse, one of the curves across the top consisted of paths turning exactly a quarter of the way round the table at each bounce, on average. On that curve every path closes after four bounces, tracing a quadrilateral inscribed in the ellipse; there is a whole circle of such quadrilaterals, one through each starting point, which is exactly the situation a twist that cannot avoid two points studied. The Poincaré–Birkhoff theorem says that when an area-preserving map is perturbed, a circle of periodic orbits like this does not survive whole: at least two of its periodic orbits persist, one stable and one unstable, and the rest of the circle is destroyed. In the bent table the stable quadrilateral sits at the centre of each red island, the unstable one sits between neighbouring islands, and every other path near the old curve is either trapped in an island, circling the stable quadrilateral, or passes outside the chain.

The second is a chaotic band. The figure-of-eight through the foci was made of paths approaching the unstable two-bounce orbit along the long axis. In the bent table those paths no longer join up smoothly, and a thin region round where they were is filled with dots that follow no curve at all. A path started there wanders up and down the band, sometimes rotating round the table and sometimes oscillating across it, and its future depends sensitively on where it began.

Rotation numbers, smooth and stepped #

The islands can be measured. Every path has a rotation number, the average fraction of a turn it advances per bounce, computed here over 3,000 bounces. The next figure plots it against the tangential component of a path’s first direction as it leaves the top of the table, for the paths that rotate.

In the ellipse the rotation number falls smoothly from nearly a half, for paths that just miss the foci, towards nought, for paths that hug the wall. Each starting direction has its own caustic, and each caustic its own rotation number, given by a ratio of elliptic integrals. A quarter is passed at one starting direction and no other.

In the bent table the curve is stepped. Over a band of starting directions about $0.06$ wide, every path has rotation number exactly a quarter, because all of them are trapped in the islands and circle the stable quadrilateral, advancing a quarter-turn per bounce on average however they started. Smaller flat stretches appear at a third and at a fifth, where narrower island chains have formed, and close to the paths through the foci the curve jumps about irregularly: those starts are in the chaotic band, and a run of 3,000 bounces gives a different average for each. The shape is the same as the staircase that the circle map builds from mode locking, and for the same reason. A rational rotation number is a resonance, and a perturbation can lock a whole interval of motions onto it.

Why a quarter is broken so much more than a third has a simple source. The bend $\varepsilon x^4$, written as a function round the table, has components that repeat twice and four times per turn, and paths that close after four bounces feel the second directly, at first order in $\varepsilon$. Paths that close after three or five bounces feel the bend only through combinations of its components, at higher order, and their islands are correspondingly narrow. A bend with threefold symmetry would break the third first.

How big the damage is #

How wide is the island chain, as a function of the bend? The next figure measures the width of the band of starting directions locked at a quarter-turn for five bends, from $\varepsilon = 0.0025$ to $0.04$, finding each edge by bisection.

The widths run from $0.018$ at the smallest bend to $0.069$ at the largest, and on logarithmic scales they lie on a line of slope $0.48$. The island chain grows like the square root of the bend. The square root is the signature of a resonance in any nearly integrable system: near a resonance the motion behaves like a pendulum whose restoring force is proportional to the perturbation, and the width of a pendulum’s region of oscillation is proportional to the square root of its restoring force. The same root appeared in how a lock comes apart, where the width of a locked plateau in the circle map grew as the square root of its distance from the edge.

The square root has a striking consequence. A bend sixteen times smaller leaves an island chain a quarter as wide, not a sixteenth. The damage done by a small bend is small, but it shrinks slowly, and there is no bend small enough to leave the quarter-turn curve intact. The ellipse is not a robust example of integrability that its neighbours approximately share. It is an isolated point, and its neighbours break, measurably, at every scale tested here.

What survives, and what was proved #

The phase portrait of the bent table was mostly smooth curves, and that is not an accident of the small bend. Vladimir Lazutkin proved in 1973 that every sufficiently smooth table with strictly positive curvature has caustics near its wall: a family of curves, filling a set of positive area in the phase portrait, along which the whispering-gallery paths are confined, with rotation numbers that are badly approximable by fractions. This is the billiard version of the Kolmogorov–Arnold–Moser theorem, the result the last circle to break traced for the standard map: under a small enough perturbation, most invariant curves survive, and the ones that break are near the resonances. John Mather showed in 1982 that the curvature condition is necessary: a table whose wall is flat at even one point has no caustics at all.

So every smooth convex table has many caustics, and the question is only whether it can have all of them. Birkhoff’s question, sharpened by Hillel Poritsky in 1950 into what is now called the Birkhoff conjecture, asks whether a table whose phase portrait near the wall is filled entirely with caustics, with no islands and no chaos, must be an ellipse. The figures here show why an answer is hard to compute: a bend can be made so small that its islands and chaotic bands are thinner than any picture can resolve, and still be there.

The partial answers are recent. Misha Bialy proved in 1993 that if the whole phase portrait, right out to the two-bounce orbits, is filled with smooth invariant curves, the table is a circle. Artur Avila, Jacopo De Simoi and Vadim Kaloshin proved in 2016 that a table close to a circle with caustics of rotation number $1/q$ for every $q$ of three or more must be an ellipse, and Kaloshin and Alfonso Sorrentino extended that in 2018 to tables close to any ellipse. In 2022 Bialy and Andrey Mironov proved the conjecture for tables with a centre of symmetry, provided the caustics reach the rotation number a quarter — the very one the bent table in this essay broke first.

Still open: whether only the ellipse keeps them all #

The Birkhoff conjecture is open. No one has proved that a smooth convex table whose paths near the wall all lie on caustics must be an ellipse, and no one has found a counterexample. The known proofs need the table to be close to an ellipse, or symmetric under a half-turn, or integrable across a large region of its phase portrait, and the general case has none of these to lean on.

The question has a sharper form in terms of the periodic paths. That the ellipse carries a whole curve of quadrilaterals, one through every starting point, is an instance of Poncelet’s porism, the nineteenth-century theorem that if one polygon is inscribed in one conic and circumscribed about another, a whole family of them is. A single such curve can be kept by tuning a bend rather than taking a random one: Yuliy Baryshnikov and Vadim Zharnitsky showed in 2006 that tables other than the ellipse can carry a whole curve of periodic paths of a given period. What nobody has found is a table other than the ellipse carrying such curves for every period at once, and whether one exists is the form of Birkhoff’s question that the proofs near the circle and the ellipse actually answer.

One table among all tables #

The ellipse’s billiard has a conserved quantity, $\lambda$, and the conserved quantity is a confocal conic. Everything orderly about its phase portrait follows from that: every path on a smooth curve, two-bounce orbits at the centres and crossings, a figure-of-eight through the foci. Bending the table by a percent and a half keeps most of that order and removes the conic. Where the quarter-turn curve was, a chain of four islands appears, a stable quadrilateral and an unstable one inside the gap the curve left; where the paths through the foci were, a band of chaos; and the islands grow as the square root of the bend, so that every bend, however slight, leaves a mark.

The confocal conics, met earlier as a coordinate grid, as the curves of Ivory’s equal diagonals and as the mirrors that send rays from focus to focus, here turn out to be the reason one shape of table is perfectly orderly. Whether they are the only possible reason is Birkhoff’s question, and after a century the answer is known only close to the ellipse itself.