Minima de L'intégrale D'action du Problème Newtoniende 4 Corps de Masses Égales Dans R3: Orbites `Hip-Hop'

Details Minima de L'intégrale D'action du Problème Newtoniende 4 Corps de Masses Égales Dans R3: Orbites `Hip-Hop' Minima de L'intégrale D'action du Problème Newtoniende 4 Corps de Masses Égales Dans R3: Orbites `Hip-Hop'

Sep 2000

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000cemda..77..139c&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, v. 77, Issue 2, p. 139-151 (2000).

Astronomy and Astrophysics

Astronomy

15

4-Body Problem, Periodic Orbit, Lagrangian Action, Homographic Solutions, 4-Body Problem, Periodic Orbit, Lagrangian Action, Homographic Solutions

Scientific paper

We consider the problem of 4 bodies of equal masses in R 3 for the Newtonian r-1 potential. We address the question of the absolute minima of the action integral among (anti)symmetric loops of class H 1 whose period is fixed. It is the simplest case for which the results of [4] (corrected in [5]) do not apply: the minima cannot be the relative equilibria whose configuration is an absolute minimum of the potential among the configurations having a given moment of inertia with respect to their center of mass. This is because the regular tetrahedron cannot have a relative equilibrium motion in R 3 (see [2]). We show that the absolute minima of the action are not homographic motions. We also show that if we force the configuration to admit a certain type of symmetry of order 4, the absolute minimum is a collisionless orbit whose configuration ‘hesitates’ between the central configuration of the square and the one of the tetrahedron. We call these orbits ‘hip-hop’. A similar result holds in case of a symmetry of order 3 where the central configuration of the equilateral triangle with a body at the center of mass replaces the square.

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