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The N-Body Problem on Coadjoint Orbits

Holger Dullin, Richard Montgomery

TL;DR

This work develops a symplectic reduction of the spatial $n$-body problem via the Galilean group to a Lax-pair formulation on $\mathfrak{sp}(2n-2)$, linking it to Albouy-Chenciner’s reduced equations. By exploiting a Howe dual-pair structure between $O(d)$ and $Sp(2n-2)$ and Xu’s normal form, the authors identify the reduced spaces at nonzero angular momentum with closures of coadjoint orbits in $\mathfrak{sp}(2n-2)^*$. In the physically relevant case $d=3$, the spatial reduced space splits into two strata: a rank-3 orbit (spatial motions) and a rank-2 planar orbit (frontier), with the planar locus acting as a singular boundary that can be resolved via $SO(3)$ reduction. The paper also shows the Lax-pair equations are equivalent to Albouy-Chenciner’s reduced system, clarifying the invariant-theoretic and geometric structure behind the reductions and providing a general framework applicable in any dimension $d$ and for all $n$. This approach yields a precise, stratified description of the reduced dynamics and connects classical celestial-mechanics reductions with modern symplectic and Poisson geometry.

Abstract

We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group $Sp(2n-2)$. The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group $O(3)$. The theorem is a special case of a general theorem (Theorem 2) which holds for the $n$-body problem in any dimension $d$. This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the $n$-body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the $n$-body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices.

The N-Body Problem on Coadjoint Orbits

TL;DR

This work develops a symplectic reduction of the spatial -body problem via the Galilean group to a Lax-pair formulation on , linking it to Albouy-Chenciner’s reduced equations. By exploiting a Howe dual-pair structure between and and Xu’s normal form, the authors identify the reduced spaces at nonzero angular momentum with closures of coadjoint orbits in . In the physically relevant case , the spatial reduced space splits into two strata: a rank-3 orbit (spatial motions) and a rank-2 planar orbit (frontier), with the planar locus acting as a singular boundary that can be resolved via reduction. The paper also shows the Lax-pair equations are equivalent to Albouy-Chenciner’s reduced system, clarifying the invariant-theoretic and geometric structure behind the reductions and providing a general framework applicable in any dimension and for all . This approach yields a precise, stratified description of the reduced dynamics and connects classical celestial-mechanics reductions with modern symplectic and Poisson geometry.

Abstract

We show (Theorem 3) that the symplectic reduction of the spatial -body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group . The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group . The theorem is a special case of a general theorem (Theorem 2) which holds for the -body problem in any dimension . This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the -body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the -body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices.

Paper Structure

This paper contains 25 sections, 12 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Forming the quotient.
  3. Angular momentum and spectral invariance
  4. Structure of Paper
  5. History and Antecedents
  6. Deriving the Lax Pair
  7. Angular Momentum and its spectrum
  8. Structure and Dual Pairs
  9. Symplectic Reduction and coadjoint orbits.
  10. The General Case and normal forms
  11. The case of bodies in 3-space
  12. Resolving the cone singularity: $SO(3)$ reduction
  13. Extraneous dimensions and dimension counts.
  14. Deriving the Quotient Map and Reduced Equations
  15. Newton's equations
  16. ...and 10 more sections

Key Result

Theorem 1

The quotient space of $n$-body phase space $P_{d, n}$ by the action of the Galilean group is the Poisson variety $symm_+(2n-2;d) \subset sp(2n-2)^* = \mathfrak g^*$ with corresponding quotient map $G \circ \pi_{trans}$. The isomorphism $\mathfrak g \to \mathfrak g^*$ takes the Lax pair equations Lax

Figures (2)

  • Figure 1: The planar locus is the singular locus for the spatial reduced spaces at non-zero angular momentum. The local structure around the planar locus is that of smooth manifold times a cone over a real projective space.
  • Figure 2: The $n=2$ Poisson structure with a symplectic leaf $|L| = const$.

Theorems & Definitions (34)

  • Remark 1
  • Theorem 1
  • Lemma 1: Spectral Lemma
  • proof
  • Remark 2
  • Remark 3
  • Definition 1
  • Lemma 2
  • proof
  • Definition 2
  • ...and 24 more