Journal
CELESTIAL MECHANICS & DYNAMICAL ASTRONOMY
Volume 77, Issue 2, Pages 139-152Publisher
KLUWER ACADEMIC PUBL
DOI: 10.1023/A:1008381001328
Keywords
4-body problem; periodic orbit; Lagrangian action; homographic solutions
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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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