Electron. J. Diff. Eqns.,
Vol. 2008(2008), No. 156, pp. 114.
Nbody problem in
:
Necessary conditions for a constant configuration measure
K. Zare
Abstract:
A formulation of the Nbody problem is presented in which
and
are the mass and the position vector of
the ith body,
and
().
The configuration measure
,
where
is the Poincare's force function,
which plays an important role in this formulation.
The orbit plane is a two dimensional linear subspace of
spanned by the position vector
and the velocity vector
.
The Nbody motion in
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
,
it is necessary that the orbit in the orbit plane be elliptic
(),
parabolic
()
or hyperbolic
()
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
.
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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