A magnetic monopole is defined (if one exists) by a magnetic field .singularity of the form
, where b is a constant (a measure of the magnetic charge, as it were). Suppose a particle of mass m moves in the field of a magnetic monopole and a central force field derived from the potential V(r) = -k/r.
a) Find the form of Newton's equation of motion, using the Lorentz force given by Eq. (1.60). By looking at the product
. Show that while the mechanical angular momentum is not conserved (the field of force is noncentral) there is a conserved vector
.
b) By paralleling the steps leading from Hq. (3.79) to Eq. (3.82), show that for some f (r) there is a conserved vector analogous to the Laplace-Runge-Lenz vector in which D plays the same role as L in the pure Kepler force problem.