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Collective branch regularization of simultaneous binary collisions in the 3D -body problem

### Abstract

In this work we study simultaneous binary collision (SBC) singularities of binaries in the three dimensional classical gravitational -body problem. We show the following: (1) In *the generalized Kustaanheimo–Stiefel variables*, the totality of SBC orbits, the totality of simultaneous binary collisions (SBE) orbits, and the collision singularity itself together form a real analytic submanifold which we call the *collision-ejection manifold*. (2) We use the collision-ejection manifold to show geometrically, without writing down any power series, that SBC solutions can be *collectively analytically continued*. That is, all SBC orbits, not just a single orbit, can be written as a convergent power series in with coefficients that depend real analytically on initial conditions that lie in a real analytic submanifold. (3) There are two important ingredients in our work. (i) We use the intrinsic energies and properly rescaled intrinsic angular momenta of the binaries as variables in order to reduce the order of the singularity and to parametrize (distinguish between different) collision orbits that constitute the stable manifolds of the rest points that appear on the collision manifold in the McGehee coordinates. (ii) We use what we call *the Kustaanheimo–Stiefel-projective transformation near a SBC singularity* to resolve the singularity and isolate collision and ejection orbits from nearby near-collision and near-ejection orbits. We will see that quaternionic multiplication and quaternionic projective spaces are not suitable.

© 2009 American Institute of Physics

Received 23 September 2008
Accepted 13 March 2009
Published online 05 May 2009

Article outline:

I. INTRODUCTION
A. The different types of regularization
B. Analytic continuation (branch regularization)
C. Binary collisions
D. Block regularization
E. Summary of the main results
F. The LCT
G. The KSM (Ref. 43)
H. The KSM versus quaternionic multiplication
II. MAIN RESULTS
A. Notation and definitions
B. Equations of motion in physical space
C. -consistent vector fields and second order equations
D. The GKS variables
III. EQUIVARIANT VECTOR FIELDS ON AND
A. The derivative of
B. The horizontal bundle
C. Vector fields on the quotient space
D. The second derivative of
E. The derivative of
F. Systems of equations on the quotient space
G. Lifting second order equation from to
H. Second order equations on the quotient space
I. The Kepler problem
IV. THE KS TRANSFORMATION FOR BINARIES
A. The spaces
B. The map
C. The fibration of
D. The principal bundle
E. Lifting vector fields to
F. Second order equations to
G. The quotient spaces
H. Second order equations on the quotient space
V. THE GKS TRANSFORMATION
A. The neighborhood
B. Equations of motion in the GKS variables
VI. THE PROJECTIVE KS TRANSFORMATION
A. A KS-projective chart
B. The KSPT
C. Transition functions for the KSPT
D. The KS-projectivized vector field(6.8) and (6.9)
E. Rest points of the KS-projectivized vector field
F. Blowing up the singularity via the KS-projective variables
G. Rest points as a graph
H. Extending the KSPT to all of
I. Blowing up the singularity via the GKS variables
J. Notation
K. Linearization of the vector field(6.8) and (6.9) at rest points
VII. THE CE MANIFOLD
A. Proof of Theorem 1, part (1)
B. Proof of Theorem 1, part (2)
C. Proof of Theorem 1, part (3)
D. Proof of Theorem 2
E. Proof of Theorem 3
F. Proof of Theorem 4
VIII. CAC
A. The existence of power series expansion for the KS variables
B. Proof of Theorem 5, part (1)
C. Proof of Theorem 5, part (2)
IX. ASYMPTOTIC BEHAVIOR OF COLLISION ORBITS

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2009-05-05

2016-09-30

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