CHAOS 5- Duhem's bull
Ah! … understanding
the movement of celestial objects!
It’s an old dream...
Some people think they can
read their destiny in the sky.
Can we predict collisions between planets?
Could gravitation eject
some planets to infinity?
Or should we rather
expect a stable movement
of the solar system?
These are delicate questions!
As always in mathematics,
when faced with a problem that is too difficult,
it is better to first look for a simpler situation.
Look at this parabolic bowl.
If we launch a ball
it is subject to its weight
and the reaction force
which compels it to remain on the surface.
We see that the problem is analogous
to that of a planet
attracted by a sun.
The movement seems too regular
to resemble a complex solar system
with many planets.
Take a bowl that is a bit more complicated.
The ball is still subject to its weight
and to the reaction force of the surface.
The movement is really complicated.
Let’s take away the weight of the ball
but keep the reaction force of the surface.
Listen to Pierre Duhem,
the philosopher of science,
presenting the work of
mathematician Hadamard,
published in 1898 in an article entitled
"On geodesic surfaces
with opposite curvatures."
A material mass
slides on a surface;
no gravity, [...]
no friction hinders its movement.
It describes a line
that geometers call
a geodesic curve
of the surface under consideration.
When we choose the initial position of
our material point
and the direction of its initial velocity,
the geodesic is completely defined.
Imagine the forehead of a bull,
with protrusions from which the horns extend [...]
and passages
between these protrusions [...]
but’s let’s extend the horns
towards infinity;
we will then have one of
the surfaces that we want to study.
Extend the horns towards infinity?
That must be a mathematician’s idea!
Launching a ball
on the forehead of a bull
with infinitely long horns…
We will first use
a different example,
but one that is in the end quite close
to Hadamard’s geodesics:
the game of billiards.
Here is a rectangular billiard table.
I shoot a ball…
…and then another,
almost in the same way.
Too simple...
The two balls follow trajectories that are very close.
An elliptical billiard table?
Still too simple!
Here also, two balls that leave
with similar initial conditions
have trajectories that remain close together.
It is as if we were
drawing geodesics on the forehead of a bull
without horns!
Let’s add a ‘horn’:
a circular stud.
Now look at two nearby trajectories.
They hit the stud
and the first goes off in one direction,
and the other in a completely different direction…
Their futures quickly become very different.
Here is an infinite plane where balls
can roll without any friction.
There are three circular obstacles:
let’s call them A, B and C.
Now here it’s as if we were observing
the geodesics on a surface with three horns.
Let’s take a ball and shoot it!
If we aim well, the trajectory can
hit A then B then C then A then B then C etc...
a periodic triangular trajectory.
Are there other periodic orbits?
Hadamard proves a beautiful theorem.
He says that if you choose any
word written with
the three letters ABC
with the condition that
consecutive letters are different,
for example ABABCABC
…then there is
a unique periodic trajectory
that visits the studs successively
in the order dictated by the word in question.
You see the complexity of the situation:
for each word,
there is a periodic trajectory.
And there are many words!
For example, the word ABCBCBCBCBC
is a trajectory that pretends
to hit only B and C,
but once every 11 rebounds
it will hit stud A!
Of course, we must aim precisely,
very precisely!
All this makes one think of the real numbers.
Some of them have
a periodic decimal notation,
and these are the rational numbers.
123/999 for example is
0.123123123…
and 2/7 is 0.285714 285714 285714... etc..
Irrational numbers have
a decimal notation
that is not periodic...
pi for example.
Our billiard table’s similar.
Some trajectories are periodic
and are described by
a periodic word in the letters A, B, C.
Others are non-periodic
and are described by an infinite word.
Still others visit A, B, C
a finite number of times, and then
go off to infinity
and never return.
Here is Duhem’ bull:
«First, there are geodesics
that close on themselves.
There are others that,
without ever coming back
to their starting point
never end up infinitely far away from it;
some keep turning
around the right horn,
and others around the left […];
other, more complicated ones,
alternate turns around one horn
with turns around the other one,
following certain rules [...]
On our bull’s head [...]
there will be geodesics
that will go to infinity,
one by climbing the right horn,
others by climbing the left horn [...].»
Imagine the Moon,
that has always been
Earth’s companion,
suddenly deciding
to shoot off towards infinity.
«Despite this complication,
if one knows with absolute accuracy
the initial position of a point
on the forehead of this bull,
and the direction of the initial velocity,
then the geodesic that this point
will follow during its movement
will be unambiguously fixed.
It will be quite different
if the initial conditions
are not known mathematically,
but practically"
Appreciate the subtlety:
not mathematically but practically!
Watch these geodesics that start
n almost the same conditions.
They follow almost the same path
for a while and then
they separate!
The green, red and blue geodesics
have completely different futures.
«lf a point is launched
on the surface in question
from a position that is given geometrically,
with a speed that is given geometrically,
then mathematical deduction
can determine the trajectory of
this point and determine
if this trajectory moves away to infinity or not.
But, for the physicist, "
this deduction is forever unusable.»
So far all of this applies to geodesics
or billiard ball trajectories.
Can this be applied
to everyday life,
for instance for the movement of planets?
The question of determining my future
is still unanswered!