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Non-integrability of the $n$-body problem

Andrzej J. Maciejewski, Maria Przybylska, Thierry Combot

TL;DR

The paper proves that the planar $n$-body problem is not integrable on a common level ${\mathscr M}_{h,c}$ of the known first integrals when $(h,c)\neq(0,0)$, for any $n>2$ and positive masses. The authors employ differential Galois theory, analyzing variational equations along a Keplerian homographic solution associated with Euler–Moulton collinear central configurations, and reduce the problem to decoupled two-dimensional subsystems. By showing that the identity component of the differential Galois group is not solvable (via two key lemmas and Kovacic-type arguments on Fuchsian equations), they obtain a strong obstruction to restricted Liouville integrability. A crucial technical step is establishing the irreducibility of the level set ${\mathscr M}_{h,c}$, enabling the Morales–Ramis–Casale framework to apply. The result extends non-integrability to arbitrary $n>2$ and masses, highlighting the inherent complexity of celestial mechanics beyond the trivial zero-energy/angular-momentum case.

Abstract

We prove that the classical planar $n$-body problem when restricted to a common level of the energy and the angular momentum is not integrable except in the case when both values of these integrals are zero. In the proof of our theorem, we use methods of differential Galois theory.

Non-integrability of the $n$-body problem

TL;DR

The paper proves that the planar -body problem is not integrable on a common level of the known first integrals when , for any and positive masses. The authors employ differential Galois theory, analyzing variational equations along a Keplerian homographic solution associated with Euler–Moulton collinear central configurations, and reduce the problem to decoupled two-dimensional subsystems. By showing that the identity component of the differential Galois group is not solvable (via two key lemmas and Kovacic-type arguments on Fuchsian equations), they obtain a strong obstruction to restricted Liouville integrability. A crucial technical step is establishing the irreducibility of the level set , enabling the Morales–Ramis–Casale framework to apply. The result extends non-integrability to arbitrary and masses, highlighting the inherent complexity of celestial mechanics beyond the trivial zero-energy/angular-momentum case.

Abstract

We prove that the classical planar -body problem when restricted to a common level of the energy and the angular momentum is not integrable except in the case when both values of these integrals are zero. In the proof of our theorem, we use methods of differential Galois theory.

Paper Structure

This paper contains 11 sections, 17 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Notion of integrability on a level of known first integrals
  3. Central configurations and homographic solutions
  4. Variational equations
  5. Proof of Theorem
  6. Case $2hc^2\neq-\mu^2$
  7. Case $2hc^2=-\mu^2$
  8. Second order differential equations
  9. Proof of Lemma \ref{['lem:main']}
  10. Proof of Lemma \ref{['lem:main0']}
  11. Proof of Lemma \ref{['lem:quad']}

Key Result

Theorem 1.1

If $(h,c)\neq (0,0)$, then the planar $n$-body problem, with $n>2$, and positive masses $m_1, \ldots, m_n$, restricted to the level ${\mathscr M}_{h,c}$, is not integrable.

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • ...and 20 more