杏吧原创

TAKE three big rocks into space and lob them in random directions. Where do
they go? This is an ancient problem in celestial mechanics, the question of how
a collection of three or more objects will move under their mutual gravitational
attraction. You have to solve it in order to predict how bodies in the Solar
System move鈥攚here the Moon will be in a million years, for example. And
the answer is . . . well, nobody really knows.

The fact that there is no solution to the 鈥渢hree-body problem鈥� means that we
are reduced to a kind of educated guess. Orbits aren鈥檛 always simple circles or
ellipses. Without a complete and precise formula that says exactly what the Moon
will do, we have to rely on approximations鈥攃omputer calculations that
predict orbits superbly in the short term, but eventually become so uncertain
that we have to shrug our shoulders and give up.

In the past two years, however, an unexpected collection of solutions to this
ancient puzzle has come tumbling out of abstract mathematical space, revealing
beautiful, looping orbits where as many as 800 bodies dance around one another.
Somewhere out there, perhaps even in our own Galaxy, troops of stars may be
following some of these weird dances.

So what is the problem with the three-body problem? It ought to be simple, as
there are just two basic ingredients, both formulated by Isaac Newton. The first
is his equation of motion, which tells you how a mass accelerates if you apply a
force to it. The second is his law of universal gravitation, which tells you how
strong the force of gravity is between two masses. That should be enough to work
out how a set of masses affect each other鈥檚 motion.

For a system with only two bodies, such as the Earth and the Sun, the problem
isn鈥檛 too hard. Newton solved it himself, showing that the path of each object
had to be a simple kind of curve called a conic section鈥攁 circle, ellipse,
parabola or hyperbola.

Shouldn鈥檛 we be able to do the same for a system of three bodies, such as the
Earth, Moon and Sun? For any given masses and starting conditions, can you
predict their trajectories? The set of equations is certainly much more complex,
because each object constantly affects how the other two are moving, but there
was no obvious reason why the problem should be impossible.

Yet it proved dauntingly difficult. Two hundred years after Newton, the
problem remained unsolved in the general case, although a few very simple
solutions were found. Swiss mathematician Leonhard Euler found the first
solution in 1765. He showed that two bodies could move around a circle or
ellipse opposite each other, with a third in the middle, all staying in a
straight line. The second solution, found by the French mathematician Joseph
Lagrange in 1772, has all three massies moving around the outside of a circle or
ellipse.

These are very contrived situations, of course. A more astronomically
plausible solution was discovered by G. W. Hill in 1877. In Hill鈥檚 scenarios,
two masses orbit each other up close in nearly circular orbits around their
centre of mass, while the third wheels around the pair at a much larger
distance.

The general solution was proving so elusive that in 1885 King Oscar II of
Sweden and Norway offered a prize of 2500 crowns to anyone who could solve it.
The prize was awarded just four years later鈥攂ut not for a solution. The
young French mathematician, Henri Poincar茅 proved that for most
arrangements of the bodies, the three-body problem is impossible to solve. He
showed that orbits that were approximate solutions behaved unpredictably,
displaying what we now call chaos. 鈥淪mall differences in the initial conditions
produce very great ones in the final phenomena,鈥� Poincar茅 wrote.
鈥淧rediction becomes impossible.鈥� This meant there could be no tidy, precise
solutions鈥攚hich makes the recently found orbits all the more surprising.
They are rare islands of order in that sea of chaos.

These new orbits were found by a time-honoured technique of mathematical
physics called the variational principle. You guess an orbit, see how well it
fits the equations, then change it a bit and see if it fits any better. Once
you鈥檝e found which way to change it to improve the fit, you can home in on an
orbit that works, step by step. It鈥檚 rather like a hiker in the mountains who
finds his way to a lake by always walking in the direction of steepest
descent.

This is the method Cristopher Moore of the Santa Fe Institute used to find
his solution, in which all three bodies have the same mass and follow one
another around a figure of eight. If you鈥檝e seen the Harlem Globetrotters
basketball team do their three-man weave, you have the idea.

The drawback to this method is that even the final solution is always
approximate. That is, it鈥檚 only a numerical solution on a computer, rather than
a precise mathematical description of the orbit which can be shown to satisfy
the equations. Mathematicians mistrust computed approximations. They like to see
logical proof. Fortunately, Richard Montgomery of the University of California,
Santa Cruz, and Alain Chenciner of the University of Paris VII and the Bureau
des Longitudes provided just such a proof for the new figure-of-eight orbit in
1999. Instead of concentrating on the bodies themselves, they looked at the
shape of the triangle created by the three bodies. This shift in perspective
made a rigorous proof easier.

Confirming the figure of eight has been like a dam bursting, says Montgomery.
Shortly afterwards, Joseph Gerver of Rutgers University in New Jersey found a
similar solution for not three but four bodies: a figure-of-eight with an extra
twist, or 鈥渟uper-eight鈥� orbit. In this configuration the four bodies also form a
parallelogram at every instant.

Then last year came a flood of solutions. Carles Sim贸 of the
University of Barcelona found scores of solutions for N bodies of equal
mass, all travelling along the same path in a plane. Maximilian Ruffert of the
University of Edinburgh found other orbits for five or more bodies. The paths
traced out are intricate and fantastic
(see Diagram), and because the masses
seem to dance along the curve, pirouetting around each other, these orbits have
been called choreographies.

Most of the choreographies are chain-like, with loops of different sizes.
Often some of the loops are folded over or nested or arranged in a circle. And
as the number of bodies increases, so too does the number and variety of
possible choreographies. Even for five bodies, Sim贸 found at least 18
different choreographies, some of them wildly complicated. He鈥檚 found long,
linear chain orbits (N = 11), figure-of-eight orbits where the masses
all follow one another around the curve (N = 19), and flower-like
curves with looping petals (N = 8 and higher). In fact, Sim贸 has
found choreographies for as many as 799 bodies鈥攁 limit imposed only by his
laptop, he says. There is such a plethora of solutions that he can鈥檛 even devise
a classification scheme.

As beautiful as these choreographic curves are, they are also fantastical,
because all the masses must be equal for them to work鈥攁 situation unlikely
to present itself in nature. Whether choreographies with unequal masses can
exist is still an open question, but Chenciner has shown that it is impossible
if the number of bodies is less than six.

Another problem with choreographies involving many masses is their
instability. A slight disturbance to the trajectory of one of the bodies will
rapidly destroy the pattern of the orbit. 鈥淚magine Sim贸鈥檚 complicated
choreographies as made of delicate crystal glass,鈥� says Ruffert. 鈥淭hey are
beautiful to behold, but any small flick will shatter them.鈥�

It鈥檚 not very surprising. The orbits Euler found can never be stable, and
Lagrange鈥檚 are stable only if one of the three masses is much greater than the
other two. What is surprising is that not all the new solutions are so
delicate.

Eternal dance

Sim贸 has shown that the figure of eight is stable. Imagine giving one
of the three bodies in the orbit a small nudge in some random direction. Instead
of the orbit disintegrating, the masses will continue their dance, following a
path very close to their original one. Another new stable solution involves two
bodies moving in propeller-shaped orbits at right angles to each other around a
common axis, with a third body circling the two like a border collie herding
sheep. This isn鈥檛 a choreography, because the objects are all on different
paths. Robert Vanderbei of Princeton University conjectures that it鈥檚 stable for
all masses, with the collie running larger and larger circles as its mass
decreases.

The discovery of stable three-body orbits means there is a chance they could
be found out in space. Douglas Heggie of Edinburgh University has simulated what
might happen when two binary star systems collide. If the two pairs of stars
approach one another from a distance with just the right positions and speeds,
three of them can join up in a figure of eight while the fourth star
escapes.

Star clusters contain many binaries, and most of these binary pairs are at
the cluster鈥檚 core. Their interactions determine how the cluster
evolves鈥攈ow many stars stay within the cluster and how many are flung
outwards. So far, though, the significance of the new orbits is uncertain.
Heggie鈥檚 computer simulations suggest that the number of figure-of-eight orbits
is somewhere between one per galaxy and one per universe. The border-collie
orbit is potentially more common and so could be more relevant to clusters, but
no one has yet worked out how often it is likely to occur.

Despite the discovery of all these remarkable celestial waltzes, there is
much left to uncover. How the stability of the simple figure-of-eight orbit
arises is still a mathematical mystery, for example. 鈥淭here is no understanding
of why the orbit is stable, from either a physical or a mathematical point of
view,鈥� says Richard Montgomery. By delving deeper into this question, he says,
mathematicians might work out another fundamental dynamical question: whether a
tiny tweak in an orbit could be the difference between a set of bodies that stay
in a tight swarm forever, and a set that can fling out one object into deep
space.

All agree that there is still much work ahead in this new flowering of
celestial mechanics. 鈥淓very piece of the N-body problem we start to
understand opens a new world of questions,鈥� says Sim贸.