Article:
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F. Rupp (2015): Static & Restricted Rigid Rotor Configurations of Three Classical
12-6-Lennard-Jones Particles, Few Body Systems, 56(2), pp. 81-105.

DOI: 10.1007/s00601-015-0958-z


Abstract:
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Motivated by the continuous search for stable geometric configurations of atom
and molecule clusters, we analyse the planar evolution of two freely movable
point particles around a third immovable one subject to the 12-6-Lennard-Jones
potential. This tailors our discussion to systems with one very heavy particle
that can be assumed to be permanently at rest in the moving reference frame for
the whole ensemble. Relating to Lennard-Jones interactions, we allow all three
point particles to take different parameters. This breaks the symmetry
conditions that are usually imposed on such systems. Through a classical
non-regularized Hamiltonian description of our restricted three particle system,
we study the existence of genuine equilibria and rigid rotor solutions around a
single axis of rotation. We prove, depending on the choice of the Lennard-Jones
parameters, that for these genuine equilibria, collinear alignments and triangular
configurations of any shape can occur. Moreover, for the discussed type of
relative equilibria a complete classification is provided by proving the
existence of rigid rotor configurations in the plane of rotation (collinear cis
and trans as well as triangle shaped configurations) and out of the plane of
rotation (triangle shaped and flag-like configurations). Furthermore, we show that
there are no further rigid rotor solutions of the underlying equations of motion.