Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 156, pp. 1-14.
Title: N-body problem in $\mathbb{R}^n$: Necessary conditions for
a constant configuration measure}
Author: K. Zare (Texas state Univ., San Marcos, TX, USA)
Abstract:
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.