Single-Particle Electrodynamics - Assassination Science

of fixed properties; or, alternatively, if we take the ultra-relativistic limit
of the constituents, then we know that we must include the Penfield–Haus
effect. Of course, only the latter course of action is of interest to us for
the purposes of this thesis. Now, it is not clear how one could modify the
Lagrangian description of the electric-current magnetic dipole to incorporate
the Penfield–Haus effect. However, since the resulting equations of motion
are so similar to the dual of the electric dipole results, we might make a guess
that the interaction Lagrangian
L ? int = µ · (B − v×E) (4.67)
might do the trick. In fact, if one computes the Euler–Lagrange equations
due to the interaction Lagrangian (4.67), one finds results in complete agreement
with those of the current loop incorporating the Penfield–Haus effect,
including the contact force µ×J. Thus, while we have only obtained it by
guesswork, it would appear that (4.67) is in fact the appropriate Lagrangian
for a fixed magnetic dipole. As with the electric dipole case, the canonical
momentum now has an extra contribution: all up, we now have
b = p + qA − d ×B + µ×E. (4.68)
The Hamiltonian likewise follows; including all moments, in the nonrelativistic
limit, we have
(b − qA + d ×B − µ×E)2
H = − d ·E − µ·B. (4.69)
2m
Finally, the principle of extended minimal coupling can be written manifestly
covariantly, for all three moments:
b = p + qA + d·F + µ· ˜F ,
or, alternatively,
p 2 = (b − qA − d·F − µ· ˜F ) 2 = m 2 .
It doesn’t come much simpler than that.
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