Single-Particle Electrodynamics - Assassination Science

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4.3 Lagrangian mechanics
In this section, we analyse the various dipoles considered in the previous
sections from the viewpoint of Lagrangian mechanics.
4.3.1 The electric dipole
It is actually quite a simple task to obtain the interaction Lagrangian for
the electric dipole of Section 4.2.1, from first principles. To do so, one need
only note that the rigid body constraints that we employed in the definition
of that model allow us [96] to simply add the interaction Lagrangians of the
two individual charges together. Starting with the interaction Lagrangian
for each electric charge,
L int = q { ϕ(z) − v(z)·A(z) } , (4.61)
and using the position and velocity expressions for each charge listed in
Section 4.2.1, one quite quickly finds that the sum of the two individual
Lagrangians yields
L int = (d ·∇)(ϕ − v·A) − ˙ d ·A. (4.62)
To convert this into a more recognisable form, we use the property, noted
in Chapter 2, that one may add a total time derivative to the Lagrangian,
without affecting its physical content. If we add the total derivative
d t (d·A)
to (4.62), perform the derivative using the product rule, and employ the
convective derivative, we immediately find
L int = d · (E + v×B). (4.63)
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