Single-Particle Electrodynamics - Assassination Science

should also be carefully noted again that the canonical momentum has no
relationship to the velocity of the particle: in particular, we have
b 2 ≠ m 2 ,
in the general case.
Finally, it must be noted that all of the results above linking the mechanical
and canonical momenta apply only to the case of an electric charge: when
we add more complicated moments, in Chapter 4, we shall have to start this
procedure all over again. In particular, the “minimal coupling” result (2.50)
is not universal. (However, we shall see that it may in fact be generalised.)
2.4.7 Conservation laws
We now briefly return to a subject touched on above, that will not be of
any practical importance in this thesis, but which is most powerful for a
number of applications: the ability to extract conservation theorems from
the Lagrangian or Hamiltonian description of a physical system.
For the case of discrete degrees of freedom q a , the Euler–Lagrange equations
(2.22)—or equivalently Hamilton’s equations (2.28)—show that, if a
generalised coördinate q a does not explicitly appear in the Lagrangian or
Hamiltonian, then the corresponding conjugate canonical momentum b a is
constant in time. This finding is not restricted to the translational degrees
of freedom of a system: it applies to any generalised coördinate of any suitably
describable system.
Similarly, for the case of continuous field degrees of freedom φ(x), one can
extract powerful conservation laws. If the Lagrangian density L is invariant
under a continuous one-parameter set of transformations, then it can be
shown (see, e.g., [145, Sec. 2.4]) that the Euler–Lagrange equations (2.36)
yield a corresponding conserved quantity; this is known as Noether’s theorem.
We shall not elaborate further on these powerful physical and analytical
tools, but will merely remind the reader that such considerations always
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