Abstract
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 ℝp with n≥2 and 1≤p≤n+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 ℝ3; we have generalized the proof to n+1 bodies in ℝp with p≤n+1. The whole proof is much more rigorous than the previous versions (Bruns, Painlevé, Forsyth, Whittaker and Hagiara). Poincaré had picked out a mistake in the proof; we have understood and developed Poincaré’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).
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Julliard-Tosel, E. Bruns' Theorem: The Proof and Some Generalizations. Celestial Mechanics and Dynamical Astronomy 76, 241–281 (2000). https://doi.org/10.1023/A:1008346516349
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DOI: https://doi.org/10.1023/A:1008346516349