A regularization of the three-body problem

Letr ₁,r ₂,r ₃ be arbitrary coordinates of the non-zero interacting mass-pointsm ₁,m ₂,m ₃ and define the distancesR ₁=|r ₁?r ₃|,R ₂=|r ₂?r ₃|,R=|r ₁?r ₂|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties:

1. The equations of motion are regular for the two-body collisionsR ₁→0 orR ₂→0.
2. Provided thatR?R ₁ orR?R ₂, the equations of motion are numerically well behaved for close triple encounters.

Although the requirementR? min (R ₁,R ₂) may involve occasional transformations to physical variables in order to re-label the particles, all integrations are performed in regularized variables. Numerical comparisons with the standard Kustaanheimo-Stiefel regularization show that the new method gives improved accuracy per integration step at no extra computing time for a variety of examples. In addition, time reversal tests indicate that critical triple encounters may now be studied with confidence. The Hamiltonian formulation has been generalized to include the case of perturbed three-body motions and it is anticipated that this procedure will lead to further improvements ofN-body calculations.