Mechanical systems with singular equilibria and the Coulomb dynamics of three charges

Abstract

We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line.