Laplace-Runge-Lenz Vector for Magnetic Monopole

[yenchin]

This is related to Goldstein Exercise 3-28. It is not really a homework problem anymore because I have solved them. But there's something bugging me so I shall post here instead of under homework problem. The Exercise reads:

A magnetic monopole has magnetic monopole B = br/r^3 where b is a constant. 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) Show that there is a conserved vector D= L - (qb/c)r/r.
(b) 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.

If you look at http://arxiv.org/abs/nlin/0504018. There is a comment right before Eq.9 where they remarked that "there is no analogue of the integral A (the LRL vector) in the Poincare and Appell problems".

Poincare problem is the pure magnetic charge case without central force (k=0) while Appell problem seems to be the one in Goldstein's exercise [of a particle moving in the field of a Newtonian center and in the field of a magnetic monopole, assuming that the center and the monopole coincide]. So my question is, did I misunderstood this somehow and Goldstein exercise is not really the Appell problem?

 

[yenchin]

Ok. I *think* I figured out. In Goldstein's exercise, he is actually asking for something easier, i.e. whether there exists a function f(r) such that dp/dt = f(r)r/r and (d/dt)(p x D) = RHS where RHS can be made into a total derivative. However the Appell problem is requiring that dp/dt has two pieces of contribution, one from Lorentz force due to the monopole and the other Keplerian central force. The piece from Lorentz force in the RHS leads to a term of the form (1/r)(d/dt)(r/r) which cannot be made into a total derivative and so an analog of the Laplace-Runge-Lenz Vector does not exist in this case. I think .