3 Motion of a single plasma particle
3.1 Motivation
As discussed in the previous chapter, Maxwellian distributions resultwhen collisions have
maximized the local entropy. Since collisions occur infrequently in hot plasmas, many im-
portant phenomena have time scales much shorter than the time required for the plasma
to relax to a Maxwellian. A collisionless model of the plasma is thus required to charac-
terize these fast phenomena. Since randomization does not occur in collisionless plasmas,
entropy is conserved and the distribution function is typically non-Maxwellian. Such a
plasma is not in thermodynamic equilibrium and so thermodynamic concepts do not in
general apply.
In Sect.2.2 it was shown that any function constructed from constants of theparticle
motion is a solution of the collisionless Vlasov equation. It is therefore worthwhile to
develop a ‘repertoire’ of constants of the motion which can then be used to construct solu-
tions to the Vlasov equation appropriate for various circumstances. Furthermore, the study
of single particle motion is a worthy endeavor because it:
(i) develops valuable intuition,
(ii) highlights unusual situations requiring special treatment,
(iii) gives valuable insight intofluid motion.
3.2 Hamilton-Lagrange formalism v. Lorentz equation
Two mathematically equivalent formalisms describe charged particle dynamics;these are
(i) the Lorentz equation
mdv
dt
=q(E+v×B) (3.1)and (ii) Hamiltonian-Lagrangian dynamics.
The two formalisms are complementary: the Lorentz equation is intuitive and suitable
for approximate methods, whereas the more abstract Hamiltonian-Lagrange formalism ex-
ploits time and space symmetries. A brief review of the Hamiltonian-Lagrangian formalism
follows, emphasizing aspects relevant to dynamics of charged particles.
The starting point is to postulate the existence of a functionL, called the Lagrangian,
which:
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