Single-Particle Electrodynamics - Assassination Science

canonical momentum:
b a ≡ ∂ ˙qa L. (2.25)
Readers who work exclusively in the field of Quantum Mechanics will no
doubt protest at the author’s choice of symbol for the canonical momentum,
running contrary to the popular choice p; but readers who work exclusively in
the field of General Relativity will without an equal amount of doubt welcome
it with open arms—the symbol p being standard notation for the mechanical
momentum; and readers who work in both fields are probably simply glad
the author realises that a notational conflict exists at all;—gladder still that
the author will scrupulously avoid any pathetic confusion between p and b in
the remaining pages of this thesis.
To summarise, the definition (2.25) states the following, according to the
author’s notation and nomenclature: the canonical momentum coördinate b a ,
conjugate to the generalised coördinate q a , is given by the derivative of the
Lagrangian L with respect to the generalised velocity ˙q a of that coördinate.
2.4.3 The Hamiltonian function
The Euler–Lagrange equations of motion (2.22) above are obtained from
the Lagrangian L, which is a function of the generalised coördinates, the
generalised velocities, and the time. We may alternatively seek a description
of the system in terms of the generalised coördinates, the canonical momenta,
and the time. It may be shown (see, e.g., [96]) that the equations of motion
of the system can be so framed, and are encapsulated in the Hamiltonian
function,
H(q a , b a , t) ≡ ˙q a b a − L(q a , ˙q a , t), (2.26)
where we are employing the Einstein summation convention of Section A.3.8.
Starting from the Euler–Lagrange equations (2.22) one then finds that the
generalised coördinates and canonical momenta evolve according to
˙q a = ∂ ba H, (2.27)
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