Stratified dynamical systems and their boundary behaviour for three bodies in space, with insight into small vibrations

Abstract

The centre-of-mass system for many particles is stratified into strata by the rotation group action. The principal stratum consists of nonlinear configurations. The collinear configurations form a lower dimensional stratum. Classical mechanics for many particles with nonlinear configurations and for those with collinear configurations are set up on the tangent or cotangent bundles over respective strata and can be reduced by the use of rotational symmetry. A question arises as to how a many-body system behaves in a neighbourhood of a collinear configuration. The system may make a vibration to bend its collinear configuration, which is a motion taking place across the boundary of the principal stratum. This paper deals with the behaviour of those boundaries for three bodies in space. The equations of motion for small vibrations as boundary behaviour at a collinear configuration will be given as a limit of those equations of motion for nonlinear configurations by means of two key facts: that the isotropy subgroup may act non-trivially on the tangent space at the collinear configuration and that vibrations take place in a constant plane in space. Further, the perturbation of small vibrations is studied by applying Moser's averaging method to show that small vibrations may give rise to finite rotations after a sufficiently large number of periods.