Ah! Understanding the movement of celestial objects. It is an old dream. Some people think they can read their destiny in the stars.

Can we predict collisions between planets? Could gravitation eject some planets to infinity? Or should we rather expect the endless stability 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.

Consider this parabolic ball. If we launch a ball, it is subject to its weight and a reaction force which keeps it on the surface.

We see that the problem is analogous to that of a planet attracted to its star. This movement seems too regular to model a complex solar system with many planets.

Let's take a ball that is a bit more complicated. The ball is still subject to its weight and to the reaction force of the surface. Now 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 the mathematician Adamard, 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. And 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? Surely this is some fancy of a mathematician to imagine 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 Adamard's geodesics. The game of billions. 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 bumper. Now look at two nearby trajectories.

They hit the bumper 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 it is 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, and so on.

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 A, B, C under the condition that consecutive letters are different for example, AB, ABC, ABC

then there is a unique periodic trajectory that visits the bumper 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 lots of words. For example, the word ABC, BC, BC, BC BC is a trajectory that pretends to hit only B and C but once every 11 rebounds, it will hit bumper A.

Of course, we must aim precisely, very very precisely.

All this makes one think of the real numbers. When we write them in decimal, some of them are periodic. These are exactly the rational numbers. As an example, 123 over 999 is 0.123123123 and so on.

While, 2 over 7 is equal to 0.285714285714285714 and so on. Irrational numbers when written in decimal are not periodic.

Pi, for example. Our billiard table is similar. Some trajectories are periodic and are described by a periodic word in the letters A, B, and C. Others are non-periodic and are described by an infinite word. Still others visit A, B, and C a finite number of times and then

they escape to infinity, never to return. Here is Duhem's book. 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.

Others, more complicated ones, alternate turns around one horn with turns around the other ones, 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 the Earth's companion, suddenly deciding to shoot off towards infinity.

Despite this complication, if one knows with an 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 begin at the same point and start almost in the same directions. They follow almost the same path for a while and then they separate. The green, red and blue geodesics have completely different futures.

If 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 or, for instance, to the movement of the planets?

The question of telling my future is still unanswered.