Single-Particle Electrodynamics - Assassination Science

canonical energy, canonical momentum and canonical angular momentum
for a closed system; and it has a much closer connection with group theory,
which is formulated completely in canonical—not mechanical—terms.
2.4.6 Electrically charged particles
We now consider the problem of obtaining the motion of a charged particle
from a Lagrangian viewpoint. The source term −J µ A µ of (2.37) suggests
that we should try the Lagrangian
L = 1 2 mv2 − q(ϕ − v·A) (2.42)
for the nonrelativistic charged particle. Computing first the canonical momentum
coördinates b conjugate to the position variables z, we find
b i ≡ ∂ vi L = mv + qA. (2.43)
Now, since the nonrelativistic quantity mv is simply the mechanical momentum,
we find the most important relation
b = p + qA, (2.44)
or, alternatively,
p = b − qA. (2.45)
The result (2.44) or (2.45) is referred to as the principle of minimal coupling.
It is clear why one most definitely needs to have distinct and unambiguous
notation and nomenclature for the mechanical and canonical momenta:
even for the simplest electrodynamical case of an electric charge, these two
quantities are not identical.
Let us now turn immediately to the Euler–Lagrange equations (2.22). For
the Lagrangian (2.42), using the result (2.44), we find
d t (p + qA) + q∇(ϕ − v·A) = 0.
58