CHAOS 3- Newton's apple
At the age of 17,
Newton wonders why
an apple falls from an apple tree
and the Moon does not fall on the Earth.
Until that time people thought
that objects on Earth, such as apples,
do not follow the same
laws as celestial bodies.
Comparing the motion of
the Moon and an apple,
and thinking that
there might be universal physical laws,
valid on Earth as well as in the skies,
was a real revolution.
Physics was dominated
by the thoughts of Aristotle.
Each object has a place of its own,
and if we move it from that place,
it will do its best to return to it..
Everything around us
seeks its natural balance.
An apple tends to go down,
since it is its nature.
The Moon orbits the Earth
because it is its nature.
Understanding falling objects
is not an easy thing.
…and by the way, cartoon heroes
have some problems
with real physics.
This is the first law
of physics in cartoons:
Any object suspended in space
remains motionless until
it becomes aware of the situation ".
According to a cartoonist at Walt Disney:
Cartoons follow
the laws of physics
unless the contrary is funnier."
Newton came up with the
"universal law of gravitation":
" Two bodies attract with a force
proportional to each of the masses,
and inversely proportional
to the square of
the distance between them."
Where does this force of attraction come from?
How can it be explained?
How can two objects
separated by billions of kilometers
attract each other instantaneously?
This mystery was not explained
until much later, at least in part,
by Albert Einstein.
All objects,
an apple
as well as the Moon,
are attracted by the Earth:
The force of gravity
affects everything that surrounds us...
Another great idea from Newton
is that forces modify speeds.
When a speed increases,
we say that the object accelerates.
So imagine an abstract apple
subjected to no other force
than its weight.
In the vicinity of an apple tree,
gravity is approximately constant.
When the apple comes loose,
its speed is zero.
But the force of gravity
changes the speed.
The apple falls faster and faster
and crashes to the ground.
Newton goes further;
he explains how to
calculate a trajectory
if we know the forces.
A ball rolling on a plane.
What will happen to it?
Nothing!
It will continue to roll
at a constant speed.
This statement that
we find in physics books
is true of course
except that it does not match
what we are used to in daily life.
A billiard ball will eventually stop
as frictional forces slow it down.
But let’s neglect these forces!
When the ball is in contact
with the ground,
the ground reacts and applies a force
in the opposite direction that
prevents the ball from piercing the ground...
..so that the total force acting on the ball is zero.
If there is no force,
there is not change in speed
and so the speed remains constant
as we have seen before..
But when the ground gives way,
the ball is subjected to
the force of gravity, its weight,
which now changes the speed
as there is no ground anymore
to counterbalance the weight.
Newton expressed all of this
in one of the most important
formulas of physics:
F=m.a
F is force,
m the mass of the object
and a is the acceleration.
The acceleration is the rate
at which the speed changes,
in other words, the derivative of the speed...
In our situation,
the force of gravity is constant,
directed downwards.
So the acceleration is constant
and directed downward.
The velocity vector increases
each second by a constant vector.
Look at the trajectory: it’s a parabola...
If we know the forces
that are applied to a system,
we know the acceleration.
If we also know the initial position
and the initial speed,
we can then predict
the future movement.
If you aim very well and if you
choose the right initial conditions,
you can get incredible motions.
An initial condition
is not just an initial position
but also an initial speed.
Look here.
Three planets, no Sun,
follow three closed trajectories.
The motion is periodic.
Here things are more interesting.
The three planets
share the same trajectory!
And here, the three planets
also follow the same trajectory,
a figure of eight.
This is planetary choreography...
It must be said that
such choreographies are real feats:
in order to make them work,
we have to aim very well,
as in billiards,
If we change the initial conditions just a little bit,
wham!,
the beautiful choreography turns to chaos.
By the way,
why doesn’t the Moon
fall on the Earth when it is
attracted by the Earth, just like an apple?
Newton's answer is amazing:
It does fall!
If we observe what happens
in the vicinity of the Earth,
the force of gravity is not constant.
It is always directed towards the center of the Earth
and is less and less strong
as we move away from it.
The Earth attracts the Moon
by the gravitational force
which influences her trajectory.
But as this is combined
with a rotation,
the Moon is constantly falling
but never reaching its goal.
If the Moon were not
subjected to any force,
she would move in a straight line
pretending not to know the Earth.
So let's take a fictitious Moon
at 384 000 km above the Earth
and let it loose.
She falls!
Second attempt:
let’s launch the same moon
with a small initial horizontal speed.
The trajectory is deviates slightly
but we have not avoided the catastrophe
With a higher initial speed the Moon
misses the Earth narrowly and
continues turning around it periodically
A higher speed yet
and the moon rotates on a circular orbit
(more or less what happens in reality).
Too high a speed and
good-bye moon!
The solar system has
eight main planets and thousands
of secondary objects,
all interacting with one another.
In the study of such as system,
the description of the initial conditions
will involve many positions
and many speeds.
And just imagine all the data necessary
to describe the motion of the atmosphere
and its countless molecules.
Studying the trajectories of a vector field
that depends on a large number of variables,
this is the real challenge
launched by the great Newton...