A conserved quantity that is not there

A star in a galaxy moves in the combined gravity of all the other stars, which near any one star is smooth and changes slowly. Its energy is conserved, and if the galaxy is symmetric about its axis, so is its angular momentum about that axis. In the 1950s astronomers measuring the velocities of stars near the Sun found that the distribution of velocities was not what those two conservation laws alone would allow. The stars behaved as if a third quantity were conserved, constraining each orbit further. Was there a third integral of motion, and if so, what was it?

Michel Hénon, then a young astronomer, and Carl Heiles, a graduate student, decided in 1963 to test the question on the simplest model they could write down, and to test it by computer. Their answer, published in 1964, was that the third integral exists in some parts of the motion and not in others, and the dividing line is the energy. Their pictures of it became one of the founding images of chaos, appearing in the same years as Lorenz’s weather model and from an entirely different direction: a system with no friction, no forcing and no randomness, governed by conservation of energy alone.

The model #

Reduce the galaxy to its essentials: a star moving in a plane, attracted towards the centre as by two springs, with a cubic correction that gives the potential the threefold symmetry of a bar or a spiral. Hénon and Heiles chose

$$V(x, y) = \tfrac12 (x^2 + y^2) + x^2 y - \tfrac13 y^3,$$

and the motion obeys Newton’s law with this potential, conserving the energy $E = \tfrac12(\dot x^2 + \dot y^2) + V(x, y)$. Without the cubic terms the two directions would be independent oscillators, each with its own energy conserved — two integrals, and every orbit regular. The cubic terms couple them. The question is whether some other quantity, more complicated than the two separate energies, is still conserved.

Near the centre the potential is a round bowl. Further out the cubic terms pull it into a triangle, and at height $1/6$ there are three passes, saddle points at $(0, 1)$ and $(\pm\sqrt3/2, -\tfrac12)$, where the level curve is three straight lines. A star with energy below $1/6$ is trapped inside its level curve for ever; above $1/6$ it can escape through a corner. So the interesting range of energies is from 0 to $1/6$.

The question was not idle. George Contopoulos had integrated orbits in models of the Milky Way’s potential in 1960 and found that they behaved as if a third integral existed, though no formula for one was known; he even constructed approximate third integrals as power series, which seemed to work. Hénon and Heiles wanted a model simple enough that, if the series converged, the integral would be undeniable, and rich enough that, if it did not, the failure would show. The cubic potential is the lowest-order coupling that has no obvious symmetry to supply an integral for free: unlike a potential depending only on distance from the centre, it does not conserve angular momentum, so energy is the only law that certainly holds.

Cutting the orbit with a plane #

The motion has four coordinates — two positions and two velocities — and conservation of energy confines it to a three-dimensional surface. That is still too much to draw. Henri Poincaré’s device is to record the orbit only when it crosses a fixed surface: here, every time it crosses the line $x = 0$ moving to the right. Each crossing is fixed by two numbers, $y$ and the $y$-momentum, since the energy then determines the $x$-momentum. A long orbit leaves a trail of points in a plane.

If a third integral exists, each orbit is confined to a two-dimensional surface inside the three-dimensional energy surface, and the section cuts that surface in a curve. So the crossings of a regular orbit lie on a smooth closed curve, and they fill it in as the orbit goes round. If no third integral exists for an orbit, nothing confines its crossings to a curve, and they can spread over an area. The section turns the abstract question into a picture: curves or dust.

The section also turns a flow into a map. Each crossing determines the next, so the motion becomes a rule taking a point of the plane to another point, and that rule preserves area, a consequence of the conservation of energy and of Liouville’s theorem. Area-preserving maps cannot have attractors, which is why the dust of a chaotic orbit does not condense onto a fractal as the Hénon map’s does, but spreads over a region of positive area. It is also why a regular orbit’s curve is filled evenly rather than spiralling inwards.

Curves, then dust #

At energy $1/12$ every orbit started from the same family of points draws a curve. The curves nest round a few centres, the stable periodic orbits, and they fill the section completely. Hénon and Heiles found the same, and the result looked like evidence that the third integral exists: the motion is as orderly as if it were integrable. At $1/8$ the picture breaks. Some curves survive, in islands round the stable orbits, but between them a single orbit can wander over a large region, and its crossings fill it with apparently random dots. Just below the escape energy almost every starting point produces dust, and only small islands of curves remain.

The orbits here are integrated by a fourth-order Runge–Kutta method with step 0.05, and the energy, which the exact motion conserves, is checked along each orbit: it stays constant to better than a millionth. The dust is not numerical error. It is the motion.

Measuring chaos one orbit at a time #

To count the chaotic orbits rather than look at them, each orbit gets a test. Start a second orbit beside it, a hundred-millionth away, and watch how their separation grows. For a regular orbit, two neighbours on nearby curves drift apart slowly, roughly in proportion to time, because their curves are traversed at slightly different rates. For a chaotic orbit the separation grows exponentially, as in the essay on how fast two orbits part, and the rate is the Lyapunov exponent.

In the figure, the separation is renormalised whenever it grows past $10^{-5}$, so that it measures the linearised growth and can be followed indefinitely. The regular orbit’s separation grows by 1.8 orders of magnitude in time 600 and flattens. The chaotic one grows by 12.2 orders, steadily on average but in fits and starts, faster when the orbit passes near the saddles and slower when it lingers near an island. An orbit is called chaotic here if its separation grows by more than four orders of magnitude in time 600.

The renormalisation is the method Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli and Jean-Marie Strelcyn made standard in 1980 for computing Lyapunov exponents: the two orbits are kept close, so that their separation measures the linearised growth rather than the size of the region, and the logarithms of the growth factors are added. Without it, two chaotic orbits would separate to the size of the region within a few hundred units of time and stop separating, and the measurement would saturate. The test is crude, since it uses one threshold and one duration, but it agrees with the pictures: the orbits it calls chaotic are the ones whose crossings fill areas.

The share of chaos #

Sixty starting points at random on the section, at each of ten energies, give the shape Hénon and Heiles found by measuring the area covered by curves. Up to energy $1/12$ not one orbit is chaotic. At $1/10$ the share is 0.12, at $1/8$ 0.58, at $0.145$ it is 0.82, and just below $1/6$ it is 0.98. The transition is not a switch at a critical energy but a steady erosion, beginning between $1/12$ and $1/10$.

For real galaxies the lesson was immediate. Stars near the centre of a galaxy, deep in its potential well, have low energy relative to the escape energy, and their orbits are mostly regular; stars further out, or in galaxies with bars and strong non-axisymmetric features, can be largely chaotic. The apparent third integral of the solar neighbourhood is the regularity of low-energy orbits, and it is not a law of the galaxy as a whole. Chaotic orbits in barred galaxies are now thought to matter for how bars form and dissolve, since chaotic stars do not support the bar’s shape the way regular orbits do.

The test can be fooled in one direction. A chaotic orbit can stick to the edge of an island for a long time, as an orbit stuck to an island found in the standard map, and during that time it separates from its neighbour no faster than a regular one. So some orbits counted as regular may be chaotic orbits observed during a quiet spell, and the share is a lower estimate.

Why the curves survive at low energy #

At low energy the star stays near the bottom of the bowl, where the cubic terms are small compared with the quadratic ones. The motion is then a small perturbation of two uncoupled oscillators, which is integrable. The theorem of Kolmogorov, Arnold and Moser, the KAM theorem, says that under a small enough perturbation of an integrable system most of the invariant curves survive, slightly deformed: those whose frequency ratios are sufficiently irrational, far from simple fractions. The curves whose ratios are close to simple fractions break up into chains of islands and thin chaotic layers, too thin to see at energy $1/12$.

The condition on the frequencies is a condition of Diophantine approximation: a ratio survives if no fraction approximates it too well. The reason is the problem of small divisors, which the conjugacy for circle maps met directly: the series that would define the third integral have denominators of the form $m\omega_1 - n\omega_2$, and near a simple ratio these are small, the terms large, and the series fails. Contopoulos’s apparent integrals were truncations of exactly these series, accurate where the frequencies were far from resonance and meaningless where they were near it.

As the energy rises the perturbation grows, more curves break, the thin chaotic layers widen and merge, and eventually one connected chaotic sea fills most of the section. It is the same story as the last circle to break in the standard map, where the kick strength plays the part of energy. In Hénon–Heiles the energy is the knob, and the third integral exists exactly on the surviving curves — which is why the stars near the Sun, with their low energies in the galaxy’s potential, behave as if it existed.

The two orbits in the plane #

Drawn as paths of the star, the difference is visible without any section. The regular orbit weaves a band with sharp inner and outer edges, a projection of the two-dimensional surface its third integral confines it to. The chaotic orbit roams over the whole triangular region its energy allows, crossing itself at every angle. Both obey the same equations with the same energy. Neither is disturbed by anything. The only difference is the starting point.

Chaos measured as a share, not a verdict #

The surprise of 1964 was that the absence of a conservation law could be seen, and that it was not a matter of all or nothing. Before then the working assumption in celestial mechanics was that systems were either integrable or ergodic, wandering everywhere on their energy surface; Hénon and Heiles showed both at once, in one system, on one energy surface, in adjacent regions. Poincaré had known that integrals generally fail to exist, from his work on the three-body problem in the 1890s, but no one had seen what the failure looks like.

The same mixture appears far from galaxies. A billiard table can be designed to have regular and chaotic regions side by side, as the mushroom-shaped table of a room that cannot be lit does, and there the boundary between them is sharp because of the table’s geometry. In Hénon–Heiles the boundary is not sharp; it is a fractal of islands within islands, and an orbit near it can spend long times almost regular before escaping into the sea. Most conservative systems are of this generic kind, and the integrable and fully chaotic ones are the exceptions. The answer — islands in a sea, with structure at every scale at the boundaries — is now known to be the typical behaviour of conservative systems, from planetary orbits to particles in accelerators.

The model also became a test case for a different question: whether a system is integrable can sometimes be decided algebraically. The Hénon–Heiles potential belongs to a family $\tfrac12(x^2 + y^2) + x^2 y + \tfrac{c}{3} y^3$, and for a few special values of $c$ — though not Hénon and Heiles’s own $c = -1$ — the system is integrable, with a third integral that can be written down. Which values those are was found by asking when the solutions, continued into complex time, have no worse singularities than poles, the Painlevé test.

Thin chaotic layers below the resolution #

The sections are finite: each orbit is followed for a few thousand units of time, and a curve drawn by a regular orbit is indistinguishable, at this resolution, from a very thin chaotic layer. KAM theory says such layers exist round every chain of islands even at energy $1/12$, where every tested orbit looked regular. The separation test is a criterion chosen for this essay, with a threshold of four orders of magnitude in time 600; a different threshold would move the share curve slightly, though not its shape. And sixty starting points per energy give shares accurate only to about a tenth.

Orbits integrated, KAM theory quoted #

Computed here: every orbit by a fourth-order Runge–Kutta method with step 0.05, with energy conserved to $10^{-6}$; the sections at 1/12, 1/8 and just below 1/6 from fixed families of starting points; the separation test for every orbit drawn; the share of chaotic orbits among sixty random starts at ten energies; the growth of separation for two chosen orbits; and the level curves of the potential, with the three saddles checked at height 1/6.

Quoted: the history of the third-integral problem and of Hénon and Heiles’s work; the KAM theorem; the integrable cases of the family and the Painlevé test.

Still open: how much of the section is chaotic #

At a given energy, what fraction of the section is occupied by chaotic orbits? Numerically it is easy to estimate, as here. Proving anything exact about it is not. It is not known whether the chaotic sea at energy $1/8$ has positive area for certain, or whether the islands and their boundaries, with islands round islands at every scale, leave a set of chaotic orbits whose area could be computed. For the related standard map it is a well-known open problem whether the chaotic region has positive area at any parameter value, and the same question for Hénon–Heiles is equally open. The pictures are among the most reproduced in dynamics, and what they show has never been proved to be what it looks like.

A law that holds only sometimes #

The Hénon–Heiles star has a third conserved quantity on the orbits that draw curves in the section and none on the orbits that draw dust, and which kind an orbit is depends on its energy and where it starts. Up to energy 1/12 every orbit tested is regular, at 1/8 more than half are chaotic, and near the escape energy almost all are. The islands that survive are the curves the KAM theorem protects, and the sea between them is where the third integral is not there.