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What will the Solar System be like in the distant future? Will Pluto
and Neptune collide? Will the Earth be thrown into a different orbit by
the combined gravitational pull from all the other planets? You might think
that the answers are easily calculated. Just program a computer with Newton’s
laws of motion, tell it the positions of the planets now, and wait while
it grinds out the future of the Solar System for the next billion years.
Right?

Wrong. With a calculation as complicated as this, the computer is almost
certain to come up with the wrong answer. It is not that the computer, or
even that the person programming it, makes mistakes. The problem arises
because computer replaces real time with a series of snapshots. Consequently,
the calculations a computer makes are not absolutely precise, so it can
only provide us with an approximate picture of what will happen in the real
world. Normally the errors are so small that they go unnoticed, but when
computers are set to work on the enormously long string of calculations
needed to simulate the movement of the planets round the Sun, tiny errors
in each step can build up to make the final result wildly inaccurate.

Confusion reigns

The errors are inevitable because the equations describing the Solar
System are so complicated that precise solutions cannot even be attempted.
Problems arise when errors build up systematically or, worse, when the errors
become chaotic. If this happens, the error in the calculated result will
not only be large, but unpredictable too. They can lead to results that
defy the laws of physics – in an extreme case the planets could spiral into
the Sun, for example, or gain energy from nowhere and spin off into space.

These errors affect every computer model although their effects go unnoticed
because calculations usually run for a short time preventing wild variations
from building up. Even a simple pendulum could start swinging like a propeller
if the simulation were left to run for long enough.

Mathematicians have discovered they can get round these problems if
the computer model is built not on the laws of motion that apply in our
familiar three-dimensional space, but on the geometrical laws of a much
larger mathematical world called symplectic space. What we understand as
movement in our space can be represented as pure geometry in the very different
world of symplectic space. This geometry provides a much more efficient
way of representing movement mathematically: while it cannot prevent the
computer from introducing errors, it ensures that, whatever the errors,
the outcome is a physically reasonable one.

The laws of geometry in symplectic space are applied through mathematical
tools known as symplectic integrators: simple formulae that a computer can
use to create reliable simulations of chaotic and complex aspects of real
systems. Symplectic integration is already helping scientists to model the
forces between tens of thousands of atoms in a crystal lattice – a system
far too complex for conventional methods to handle reliably – and successfully
predict properties of the material such as its strength or the way it vibrates.

Not every system in our Universe is symplectic, however. Dissipative
forces such as friction or viscosity do not obey geometric laws and cannot
be simulated using symplectic integration. Applying the method to weather
forecasting is complicated, for example, because air resistance is a dissipative
force and has to be ignored.

One example where symplectic simulations can help is in designing circular
particle accelerators. The Main Ring accelerator at Fermilab in Illinois,
which was built in 1972 before symplectic motion was understood, cannot
store particles for long because small variations in their paths rapidly
build up into uncontrollably large deviations. Physicists now realise that
with the new symplectic methods they would have been able to simulate these
effects, and design the machine to cope.

Conventional computer models use position and time to deduce velocity,
but position and velocity are treated on an equal footing in symplectic
space. For instance, in a model of the Solar System each planet is defined
by six dimensions – three for its velocity in each direction and three for
its position. The dimensions are coupled by special kinds of angles known
as symplectic angles. These angles cannot be measured with a protractor:
two lines superimposed on each other, for example, have a normal angle between
them of 0 degree but a symplectic angle of 90 degree.

The special features of this weird space are its laws of geometry. That
space and geometry are closely linked can be seen by looking at the properties
of a triangle in two different types of space. We are taught at school that
the angles of a triangle add up to 180 degree, but this is not always true
if the triangle is not on a flat plane. For example, imagine a triangle
on the surface of the globe, which has one corner at the North Pole and
the other two on the equator. No matter what the angle at the pole, both
the angles at the equator will be exactly 90 degree, so the three angles
are bound to add up to more than 180 degree. By choosing the space carefully,
mathematicians can arrange for certain geometric laws to hold true. In the
example of the triangle, the angles add up to 180 degree only if the space
is flat.

If the space is complex, then the geometric laws can be complex too.
As the planets move in symplectic space acc-ording to the symplectic laws
of geo-metry, so they move in three-dimensional space according to Newton’s
laws of motion. Symplectic geometry ensures that symplectic angles remain
the same as the planets move. This geometry can describe all the complicated
motion of the Solar System.

Symplectic space has been hard to explore because its geometry is so
unlike that of three-dimensional space (the name comes from the Greek word
symplegma, meaning tangled or plaited). After many decades of study, the
breakthrough came during the 1960s when the Russian mathematician Vladimir
Arnol’d at the Moscow State University, along with Andrei Kolmogorov and
Jurgen Moser, proved a theorem that explains some of the implications that
these hidden geometrical laws hold for real motion.

In the case of a single planet in a circular orbit around the Sun, the
motion is nonchaotic and well understood. But one problem that could not
be by conventional means is what happens when there is a second planet exerting
a small gravitational pull on the first. Does the circular orbit merely
become slightly elliptical? Does it develop a chaotic wobble? Or will the
first planet wander off entirely? The Kolmogorov-Arnol’d-Moser (KAM) team
proved that all three are possible. Given certain initial conditions, the
orbit can still be regular. But if the starting conditions are slightly
different, it could be chaotic. Once in a chaotic state, the orbit might
even ‘leak out’ in a process known as Arnol’d diffusion, which causes the
planet to wander away from its circular orbit.

Alternative orbits

The problem lies in determining which type of orbit a system will adopt.
The KAM theorem shows that chaos and order are infinitely mixed. Between
any two regular orbits lie chaotic ones, and the planet could adopt any
one of an infinite number of each type of orbit. But a planet that behaves
nonchaotically can never become chaotic.

Traditional computer models of planetary orbits produce outrageous results
because the build-up of errors in the calculation leads to results that
run counter to the laws of motion on which the model is based. Symplectic
integration avoids these pitfalls by modelling not just the forces and accelerations
as happens in conventional computer simulations, but by also keeping symplectic
angles fixed in symplectic space. While the computation will still, inevitably,
accumulate errors, the KAM theorem guarantees that they will not nudge a
planet into an impossible orbit. The result does not predict the exact
motion of our Solar System, but it does provide useful information about
it. For example, astronomers can work out Pluto’s distance from the Sun
in a billion years’ time, but not which side of the Sun it will then be
on.

Although scientists have started to use symplectic integration only
recently, it is not a new idea. In the 1950s Rene de Vogalaere, a mathematician
then at the University of Notre Dame in Indiana, suggested rewriting formulae
to preserve symplectic angles at each step. But his paper was rejected by
a mathematical journal and the idea was forgotten. In 1983, it was rediscovered
independently by Ronald Ruth at Lawrence Livermore National Laboratory in
California and Feng Kang at the Chinese Academy of Sciences in Peking.

Since then, the study of symplectic integration has gone from strength
to strength and the technique is giving scientists new insights into the
workings of the Solar System. That chaotic motion exists in the Solar System
was first suggested in 1988 by conventional computer calculations. But
these were painfully and unnecessarily slow, and symplectic integrations
can be carried out in a fraction of the time to simulate this chaos far
into the future. ÐÓ°ÉÔ­´´s can now model the entire lifespan of the Solar
System, thought to be about 10 billion years, and know that the model is
reliable because it obeys the laws of symplectic geometry.

These models demonstrate vividly how chaotic the Solar System’s behaviour
is. Imagine a model of the Solar System made up of little coloured lights
zipping around the Sun in elliptical orbits. Now speed up time until the
points of light appear to be smeared into nine elliptical rings, with Neptune’s
orbit overlapping Pluto’s. The first thing you notice is that the planets’
perihelia – the points at which their elliptical orbits are closest to the
Sun – are not fixed, but rotate slowly. This phenomenon, which is known
as perihelion precession, has been known for around a century. The Earth’s
ellipse, for example, takes 112 000 years to rotate once – the sort of
period that conventional computer methods can easily simulate.

Now speed up time even more, so that a million years pass every second.
At this rate, Pluto orbits the Sun 4000 times each second and conventional
computer methods start to fail within a few seconds. The measure of the
shape of a planet’s orbit is its eccentricity – the narrower the ellipse,
the greater the eccentricity – and Pluto has the most eccentric orbit of
all the planets. Symplectic integration can show how the cumulative effects
of the gravitational pull of each planet on every other planet begin to
show up in changes in the shape of their orbits. The eccentricity of Pluto’s
orbit begins to change erratically over the next 70 million years, leading
to a 500-million-kilometre change in its maximum distance from the Sun.

Pluto’s perihelion also behaves strangely. Instead of steadily rotating,
as other planets’ perihelia do, it comes to a halt, reverses, and finally
swings back and forth by an amount that varies every 34 million years. In
addition, the tilt of its orbit relative to the Earth’s, oscillates chaotically
between 14 degree and 17 degree, like a spinning coin coming to rest. Jack
Wisdom of the Massachusetts Institute of Technology, who calculated these
figures, says the diversity of Pluto’s motion seems to be inexhaustible.

So will Pluto collide with Neptune in a billion years’ time? Probably
not. Surprisingly, the calculations clearly show the planets avoiding each
other – a phenomenon that has yet to be explained. The Solar System may
behave chaotically, but it seems destined never to look very different from
the way it is now.

Robert McLachlan is an applied mathematician at Massey University, New
Zealand. Further reading ‘The symplectic revolution: A new suppler geometry
is transforming physics and mathematics’, The Sciences, vol 30 p 28. Newton’s
Clock: Chaos in the Solar System, by Ivars Peterson, W. H. Freeman 1993

* * *

Symplectic map-making

The formulae used in symplectic calculations are very simple, the simplest
of all being the ‘standard map’. A ‘map’ is simply mathematical jargon for
a rule that shows how to move from one point to another. A computer given
the coordinates of a point can use the standard map to generate a new point.

If the first point has coordinates x and y, the formula for generating
the new coordinates is:

x_new = decimal part of (x + y)

y_new = y – sin (2&pgr;x_new),

where ‘decimal part’ means the numbers to the right of the decimal
point.

As successive points are plotted on the same plane, a picture gradually
builds up. Some starting points lead to a regular, repetitive cycle of points
which eventually fill out to form a smooth curve. For others, the point
bounces around seemingly at random. On the map shown above, different starting
points are represented by different colours. Researchers have repeated these
calculations with different starting points over a million billion (1015)
times to find the hidden patterns in this symplectic chaos.

The standard map turns out to be a mathematical model of a simple forced
pendulum. Such models do not reproduce all the aspects of a real system
but distil certain features that can then be studied. For example, the motion
of an electron travelling round a circular particle accelerator can be described
by the same model. In this case, the axes of the picture show the position
and velocity of the particle. Different starting positions correspond to
particles in different orbits, and the points on the picture are snapshots
of the velocity and position of the particle at regular intervals. The
orbits that remain stable produce a repetitive cycle of points, while chaotic
orbits produce a random selection of points. Physicists use simulations
like this when designing particle accelerators to determine how the beam
will behave and how it must be controlled.