The three bodies problem and Lagrange points

Now that I have tired of studying I want to talk about some very funny points in the orbits of bodies telling, for you to understand, a bit of history (as we are).

As you know, Newton shed his laws by studying the orbits of bodies. Hence Newtonian mechanics is based, mainly, on the position of objects and their modification. It is simple.

Newton’s studies were based primarily on the study of the two bodies, the forces between them and the trajectories. Thanks to him, he calculated with great precision the orbits of the planets with respect to the Sun.

But, friends, the solar system is not only two bodies, but there are many more. That is why, after studying the movement of two bodies, physicists and mathematicians went on to the next challenge: the problem of the three bodies.

Although the problem of the two bodies through differential equations has a solution (let there be an integral solution – you know, the integral is the natural enemy of the differential), the problem of the three bodies can not be solved because of Initial conditions since there is no formula to govern it (as Poincaré showed). Let’s say that the differential equations do not have an exact solution, although if you can make approximations to the solution using other methods (anyone who knows about differential equations is sure to sound the series approximations … that’s where the subject is).

Where it has a solution is what is called “the problem of three bodies restricted” which is a “sub version” of the problem of three bodies where one of them has a mass that practically does not count, being practically a problem Of the two bodies.

The solution method is not based so much on the positions and their calculation and if on the energies, hence Lagrange (remember that we studied Lagrangian mechanics) was the master in solving this problem.

Thanks to Lagrange and its solution (to circular orbits) some points were found in the orbits that are very used, Lagrange points.

The Lagrange points are zones in the orbit between two objects where, the third, of despicable mass does not move. That is, they are points where the force acting on the object of negligible mass gives, as a result, a geosynchronous orbit. Let’s say that if we have 3 objects M, m and m ‘where the mass of M >> m >>> m’, the orbit of m ‘is kept at constant distance from the other two.

This, which may seem not only stupid but also a curiosity defines a total of 5 very important points to place objects that need to know that they will always be there … for example, satellites with which we want, I do not know, For example, to measure things of the object of mass M or of object of mass m (above-mentioned). In fact, at these points are located (currently) satellites of observation of the solar chromosphere and the terrestrial magnetic field.

In addition, in these points and because the object has to have a very small mass with respect to the others, they are zones of concentration of asteroids that, rotate with the planets (L4 and L5), having the particularity that the mass of the object That is there can be … “very large”, thanks to how the forces interact. In these two points there are what are called “Trojan” bodies, and all the planets in the solar system have asteroids at those points, all of them. For example, Jupiter, because it is a gravitational mastodon has more than 8000 objects there, so quiet, while the Earth has only one, called 2010 TK7.

More info: NASA