# The Integral Manifolds of the Three Body Problem

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*Christopher K. McCord; Kenneth R. Meyer; Quidong Wang*

The phase space of the spatial three-body problem is an open subset in \({\mathbb R}^{18}\). Holding the ten classical integrals of energy, center of mass, linear and angular momentum fixed defines an eight dimensional submanifold. For fixed nonzero angular momentum, the topology of this manifold depends only on the energy. This volume computes the homology of this manifold for all energy values. This table of homology shows that for negative energy, the integral manifolds undergo seven bifurcations. Four of these are the well-known bifurcations due to central configurations, and three are due to “critical points at infinity”. This disproves Birkhoff's conjecture that the bifurcations occur only at central configurations.

#### Table of Contents

# Table of Contents

## The Integral Manifolds of the Three Body Problem

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. The Decomposition of the Spaces 1120
- Chapter 3. The Cohomology 2736
- Chapter 4. The analysis of k(c,h) for equal masses 6170
- 1. y[sub(1)][sup(2)] + y[sub(2)][sup(2)] as function of σ,τ for equal masses 6170
- 2. The semi-minor axis of the ellipse for equal masses 6372
- 3. The graphs of Z = f(X) and Z = g(X) for equal masses 6574
- 4. The semi- major axis of the ellipse for equal masses 6978
- 5. The feasible region c(c, h) 7079
- 6. k[sub(R)](c,h) for equal masses 7180
- 7. Orientation in k(c,h) 7281
- 8. Positive energy 7584

- Chapter 5. The analysis of k(c,h) for general masses 7786
- Bibliography 91100