n-body Problem - Sundman's Theorem For The 3-body Problem

Sundman's Theorem For The 3-body Problem

In 1912, the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of for the 3-body problem. This series is convergent for all real t, except initial data which correspond to zero angular momentum. However these initial data are not generic since they have Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed in the next section, the only singularities in the 3-body problem are

1. binary collisions
2. triple collisions.

Now collisions, whether binary or triple (in fact of arbitrary order), are somehow improbable since it has been shown that they correspond to a set of initial data of measure zero. However there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

1. He first was able, using an appropriate change of variables, to continue analytically the solution beyond the binary collision, in a process known as regularization.
2. He then proved that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to he removed all real singularities from the transformed equations for the 3-body problem.
3. The next step consisted in showing that if then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using the Cauchy existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis.
4. The last step is then to find a conformal transformation which maps this strip into the unit disc. For example if (the new variable after the regularization) and if then this map is given by

This finishes the proof of Sundman's theorem. Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms that his solution is of little practical use.