Confined free motion under a dipole potential

Abstract

The classical motion of a particle in a dipolar potential, $U_{\hbox{dip}}(q) = - {k}/{q^2}$, and free motion along a curve in phase space are proven to be equivalent. We also prove that the singularity at $q=0$ in the dipolar potential is strong enough as to prevent the flow of particles from one side of the singularity to the other. This effect does not depende on whether the dipole potential is regarded as attractive ($k>0$) or as repulsive ($k<0$). All the proofs are given using the Hamitonian formalism, therefore they may be used for illustrating the power the Hamiltonian approach may confer in analysing different mecanical systems. The discussion is keep within the reach of advanced undergraduate or graduate students of Hamiltonian mechanics.