We investigate three-body motion in three dimensions under the interaction
potential proportional to r^alpha (alpha 0) or log r, where r represents
the mutual distance between bodies, with the following conditions: (I) the
moment of inertia is non-zero constant, (II) the angular momentum is zero, and
(III) one body is on the centre of mass at an instant.
We prove that the motion which satisfies conditions (I)-(III) with equal
masses for alpha -2, 2, 4 is impossible. And motions which satisfy the
same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits
are not figure-eight and these motions have collision. Therefore
non-conservation of the moment of inertia for figure-eight choreography for
alpha -2 is proved.
We also prove that the motion which satisfies conditions (I)-(III) with
general masses under the Newtonian potential alpha=-1 is impossible.
%0 Generic
%1 citeulike:46673
%A Fujiwara, Toshiaki
%A Fukuda, Hiroshi
%A Ozaki, Hiroshi
%D 2003
%K choreography evolution inertia moment
%T Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography
%U http://arxiv.org/abs/math-ph/0304014
%X We investigate three-body motion in three dimensions under the interaction
potential proportional to r^alpha (alpha 0) or log r, where r represents
the mutual distance between bodies, with the following conditions: (I) the
moment of inertia is non-zero constant, (II) the angular momentum is zero, and
(III) one body is on the centre of mass at an instant.
We prove that the motion which satisfies conditions (I)-(III) with equal
masses for alpha -2, 2, 4 is impossible. And motions which satisfy the
same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits
are not figure-eight and these motions have collision. Therefore
non-conservation of the moment of inertia for figure-eight choreography for
alpha -2 is proved.
We also prove that the motion which satisfies conditions (I)-(III) with
general masses under the Newtonian potential alpha=-1 is impossible.
@misc{citeulike:46673,
abstract = {We investigate three-body motion in three dimensions under the interaction
potential proportional to r^alpha (alpha \neq 0) or log r, where r represents
the mutual distance between bodies, with the following conditions: (I) the
moment of inertia is non-zero constant, (II) the angular momentum is zero, and
(III) one body is on the centre of mass at an instant.
We prove that the motion which satisfies conditions (I)-(III) with equal
masses for alpha \neq -2, 2, 4 is impossible. And motions which satisfy the
same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits
are not figure-eight and these motions have collision. Therefore
non-conservation of the moment of inertia for figure-eight choreography for
alpha \neq -2 is proved.
We also prove that the motion which satisfies conditions (I)-(III) with
general masses under the Newtonian potential alpha=-1 is impossible.},
added-at = {2007-08-18T13:22:24.000+0200},
author = {Fujiwara, Toshiaki and Fukuda, Hiroshi and Ozaki, Hiroshi},
biburl = {https://www.bibsonomy.org/bibtex/26c2823552a732fff053df04c3ba9a810/a_olympia},
citeulike-article-id = {46673},
description = {citeulike},
eprint = {math-ph/0304014},
interhash = {2bcdd185809dc2387b2479d7b67c96e7},
intrahash = {6c2823552a732fff053df04c3ba9a810},
keywords = {choreography evolution inertia moment},
month = {September},
timestamp = {2007-08-18T13:22:59.000+0200},
title = {Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography},
url = {http://arxiv.org/abs/math-ph/0304014},
year = 2003
}