Single-Particle Electrodynamics - Assassination Science

Let us return to the electric charge Lagrangian (2.42). We can obtain a
Hamiltonian description of the system by again using the result (2.26):
H ≡ v·b − L
{ }
1
= v · (mv + qA) −
2 mv2 − q(ϕ − v·A)
= 1 2 mv2 + qϕ.
We again must convert the functional dependence of this result, from being
in terms of the generalised velocity components v, to being in terms of
the canonical momentum coördinates b.
substitution, yielding
H(z, b, t) =
(b − qA)2
2m
We can use (2.43) to effect this
+ qϕ. (2.47)
It may be seen why the “minimal coupling” result (2.44) was a guiding light
in the early days of electrodynamics: not only does one need to add a scalar
potential qϕ to the free-particle Hamiltonian, one must additionally modify
the factor b 2 /2m into (b − qA) 2 /2m. Of course, this term is still simply the
nonrelativistic kinetic energy of the particle:
(b − qA) 2
2m
= p2
2m ≡ 1 2 mv2 ;
it is simply that the functional dependence of this term on the canonical
momentum changes, because the canonical momentum itself is no longer
the mechanical momentum.
The situation may be clarified even further by considering the problem
when generalised to Einsteinian mechanics. The form of the Hamiltonian
(2.47), which we can rewrite
H − qϕ =
(b − qA)2
2m ,
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