Stable compact motions of a particle driven by a central force in six-dimensional spacetime

We give a counterexample to the well-known Ehrenfest’s assertion that the existence of stable electromagnetic bound systems is impossible in spaces of more than three dimensions. If we require that the Maxwellian laws of electromagnetism be preserved for any even spacetime dimension, and that the dynamics as a whole be consistent, then the laws of mechanics must be amended by the addition of terms with higher derivatives. We consider a nonrelativistic particle with an acceleration-dependent Lagrangian which moves in an attractive 1/r31/r3 potential in five-dimensional space. There are compactly supported motions whose projections on the SO(5)(5)-reduced Hamiltonian system are Poisson equilibrium points. The nonlinearly stable equilibria correspond to physically stable motions over the direct product of two three-spheres in configuration space. The Energy-Casimir method turns out to be not appropriate for checking the stability. The studied system is shown to be stable through an analysis of numerical solutions to the equations of motion for small perturbations on the reduced phase space. This implies that falling to the center is prevented.