Electron. J. Diff. Eqns., Vol. 2008(2008), No. 156, pp. 1-14.

N-body problem in $\mathbb{R}^n$: Necessary conditions for a constant configuration measure

K. Zare

A formulation of the N-body problem is presented in which $m_{i}$ and $r_{i}\in \mathbb{R}^d$ are the mass and the position vector of the i-th body, $x=(\sqrt{m_{1}}r_{1},\dots ,\sqrt{m_{N}}r_{N})\in \mathbb{R}^n$ and $n=dN$ ($d=1,2,3$). The configuration measure $Z =|x|F$, where $F$ is the Poincare's force function, which plays an important role in this formulation. The orbit plane is a two dimensional linear subspace of $\mathbb{R}^n$ spanned by the position vector $x$ and the velocity vector $\dot{x}$. The N-body motion in $\mathbb{R}^n$ has been decomposed into an orbit in the orbit plane and the instantaneous orientation of the orbit plane. For a solution to stay on a level manifold of $Z$, it is necessary that the orbit in the orbit plane be elliptic ($h<0$), parabolic ($h=0$) or hyperbolic ($h>0$) where h is the total energy. The instantaneous orientation of the orbit plane can be obtained by integration of certain differential equations. These possible solutions include the central configuration solutions in which the orbit plane is fixed in $\mathbb{R}^n$.

Submitted June 4, 2008. Published November 20, 2008.
Math Subject Classifications: 70F10.
Key Words: Configuration measure; Saari's conjecture; central configurations; generalized vector product; generalized momentum and eccentricity vectors.

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Kal Zare
Department of Mathematics
Texas state University
601 University Drive
San Marcos, TX 78666, USA
email: kz11@txstate.edu

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