A magnetic monopole is defined (if one exists) by a magnetic field singularity of the form B=br/r3, 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 r×p˙ show that while the mechanical angular momentum is not conserved (the field of force is noncentral) there is a conserved vector
D=L−cqbrr.(b) By paralleling the steps leading from Eq. (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.