Bruns' theorem: The proof and some generalizations

We give here a proof of Bruns' Theorem which is both complete and as general as possible: Generalized Bruns' Theorem. In the Newtonian (n+1)-body problem in R-P with n greater than or equal to2 and 1 less than or equal top less than or equal ton+1, every first integral which is algebraic with respect to positions, linear momenta and time, is an algebraic function of the classical first integrals: the energy, the p(p-1)/2 components of angular momentum and the 2p integrals that come from the uniform linear motion of the center of mass. Bruns' Theorem only dealt with the Newtonian three-body problem in R-3; we have generalized the proof to n+1 bodies in R-p with p less than or equal ton+1. The whole proof is much more rigorous than the previous versions (Bruns, Painleve, Forsyth, Whittaker and Hagiara). Poincare had picked out a mistake in the proof; we have understood and developed Poincare's instructions in order to correct this point (see Subsection 3.1). We have added a new paragraph on time dependence which fills in an up to now unnoticed mistake (see Section 6). We also wrote a complete proof of a relation which was wrongly considered as obvious (see Section 3.3). Lastly, the generalization, obvious in some parts, sometimes needed significant modifications, especially for the case p=1 (see Section 4).