Confined free motion under a dipole potential

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.

Carter constant and angular momentum

We investigate the Carter-like constant in the case of a particle moving in a nonrelativistic dipolar potential. This special case is a missing link between the Carter constant in stationary and axially symmetric spacetimes (SASS) such as Kerr solution and its possible Newtonian counterpart. We use this system to carry over the definition of angular momentum from the Newtonian mechanics to the relativistic SASS.

On the theory of the nonlinear effect in polar liquids

It is shown that in order to relate the permittivity ε = ε0[1–λE2] of a nonlinear dielectric with molecular parameters using Onsager's formalism, the condition to be used is that the average value of the dipolar potential outside the cavity and with an applied field must vanish. Consequently, the potential for a rigid dipole in a spherical cavity filled up with a continuum ε∞ and surrounded by a nonlinear dielectric immersed in a strong applied field are calculated both inside and outside. The average value of the external dipole moment is deduced from the torque exerted by the field outside the cavity on the sources of the applied field.The expression thus obtained for the parameter λ leads to values that are always positive and slightly smaller than those corresponding to earlier studies. The expression deduced for the permittivity in weak fields ε0 introduces a correction to Onsager's equation consisting of a factor multiplying the square of the molecular moment. In contradiction with recent results by other authors, this factor is always less than 1 and its value is of the order of 0.9.