CHAOS 2-The Lego race
On October 26, 1676
Isaac Newton writes a letter
to his great rival Leibniz.
He wants to talk about his greatest discovery
but at the same time
he wants to keep it secret,
so he sends a riddle:
«6a, 2c, d, ae, 13e, 2f, 7i, 3l, 9n, 4o, 4q, 2r, 4s, 8t, 12v, x».
Leibniz did not decode this message
but science historians have
eventually decrypted
the Latin phrase that was hiding in the anagram:
«Data aequatione quotcunque
fluentes quantitates involvente,
fluxiones invenire et vice versa.»
In English:
«Given an equation
in any number of variables,
find its derivatives, and vice versa.»
This is calculus:
a crystal ball that is incredibly effective
at predicting the future.
It lets you calculate
the future movement of a system
when you know its present state
and the forces that affect it.
It is about solving a differential equation
and finding solutions.
Let’s look at this in the world of lego people.
After all, the mathematical world
is a bit like a game,
where everything is simpler than in real life;
a world that is sometimes childish.
The footprints indicate
the paths followed
by the athletes,
their trajectories.
Suppose that our little men
take steps at a steady pace
but that these steps are sometimes long
and sometimes short.
When the Lego goes fast,
two consecutive footsteps are far apart.
When they go slowly they are close together.
The arrow joining two consecutive steps
indicates the speed:
it is the difference
between two positions
at two consecutive moments,
hence the word "differential calculus".
The hundred meters race.
Who will win?
9'58 ", great!
Now, movement
rarely consists of jerky steps.
Look at these moving racecars:
They roll, they don’t take steps!
What do we then mean by speed?
One idea is to say that after all
this film consists of
25 frames per second
so that one can indeed think that
the motorbike is taking 25 "steps" every second!
and we can talk about its speed, just
like we were talking about the speed of the lego runners.
Newton approximates any continuous movement
as a sequence
of stepwise movements,
but with steps that become
so small that they become invisible, like in the movies.
Calculating the speed of a motion,
is called calculating a derivative.
It is the goal of "differential calculus".
Here are some movements.
These arrows indicate the speed,
and mathematicians call them vectors.
Now imagine the opposite problem.
You see arrows
drawn across the floor.
We call this a vector field.
Imagine a field of wheat ...
but instead of wheat stalks
you have a vector.
The lego people’s mission
is to move with a speed
indicated by the vector field.
Easy you say!
They look under their feet,
they see a vector
which tells them their speed,
and then they set off in that direction
at this speed.
A brief moment later
they have arrived at a new point
with a new speed,
then off they go in the new direction
and they do this over and over again
Walking is not difficult
Just put one foot in front of the other
and do it again ! "
Actually, we should explain
what we mean by
"a brief moment later!"
Newton’s answer would be
«An infinitely brief moment! ..»
We already saw that
a continuous movement is not the same thing as
a succession of steps:
it’s what you get,
when the steps become smaller and smaller.
So rather than hopping like a lego person,
take a car that rides continuously.
You will follow what is
called a trajectory;
a curve that is tangent to the vector field everywhere.
Here you have a vector field in the plane,
and two points that are
the initial positions of two motorcycles.
The theorem of Cauchy-Lipschitz
summarizes the concept of determinism:
it claims that these points
determine the future trajectories.
Starting from each point,
there is a unique trajectory
whose initial position is the given point.
Each point has its destiny,
different for everyone ...
The lego man facing his fate!
All he can do is follow his trajectory,
and two trajectories
can never cross each other.
Determining the trajectory
from the knowledge
of the velocity field
is the work of integral calculus,
which thus goes in the opposite
direction as differential calculus ...
And here is a group of lego people,
neatly ordered
like little soldiers, ready to start walking.
Off they go.
You see that the nice
original order gets broken.
It is sometimes said that
one determines the flow of a field,
as if these characters
were floating on a river,
each following their course in the current.
Think of the flow of humanity,
these seven billion lego people moving all over the Earth,
or think of the flow of billions of billions
of billions of molecules in the Earth's atmosphere.
Here is a simple example,
almost naive,
which will show us a weakness of determinism ...
Watch this vector field.
The figures move forward and as you can see,
those on the left of the central line
turn to the left
and those on the right
turn to the right.
In a way determinism is valid:
everyone follows his destiny
over which he has no control.
But on the other hand,
two men very close to each other
have very different destinies.
A little thing
can completely change the future.
In his little book "Matter and Motion",
published in 1876,
the physicist Maxwell stresses
the sensitivity of physical
phenomena to initial conditions.
Here is what he writes:
«There is a maxim…that
the same causes will always produce the same effects [...]
There is another maxim
which must not be confounded with the first,
which asserts that
“like causes produce like effects”.
This is only true when
small variations in the initial
circumstances produce only
small variations in the
final state of the system.
In a great many physical phenomena
this condition is satisfied; but there are other
cases in which a small initial variation may
produce a very great change in the final state of the system. "
For instance, a small difference
in the speed of the car
may cause an accident.
This dependence of
the future on initial conditions
is only one aspect of chaos!
But there are much more complex situations ...
Imagine for example a vector field
which is no longer drawn on the ground,
but in space.
This one for instance
that you see on a vertical plane
moving back and forth.
Now our legos don’t walk:
they fly their spaceships!
At every moment, their speed
is determined by the vector field.
See what happens to our unfortunate legos!
This is much more chaotic!
Imagine a lego person as a soothsayer:
impossible with such a roller coaster:
His predictions would be mere deceptions.
Where will he be in an hour?
No one knows!
If it is difficult to predict the future of a lego
imagine predicting the future of a human being!