Fundamentals of Plasma Physics

3.2 Hamilton-Lagrange formalism v. Lorentz equation 65

be used as the canonical coordinates in which case Eq.(3.8) gives the canonical momentum
as

P=mv+qA(x,t). (3.13)

The left hand side of Eq.(3.7) becomes

dP

dt

=m

dv

dt

+q

(

∂A

∂t

+v·∇A

)

(3.14)

while the right hand side of Eq.(3.7) becomes

∂L

∂x

= q∇(v·A)−q∇φ=q(v·∇A+v×∇×A)−q∇φ

= q(v·∇A+v×B)−q∇φ.

(3.15)

Equating the above two expressions gives the Lorentz equation where the electric field is
defined asE=−∂A/∂t−∇φin accord with Faraday’s law. This proves that Eq.(3.12)
is mathematically equivalent to the Lorentz equation when used with the principle of least
action.
The Hamiltonian associated with this Lagrangian is, in Cartesian coordinates,

H = P·v−L

=

mv^2

2

+qφ

=

(P−qA(x,t))^2

2 m

+qφ(x,t) (3.16)

where the last line is the form more suitable for use with Hamilton’s equations, i.e.,H=
H(x,P,t). Equation (3.16) also shows thatHis, as promised, the particle energy. If
generalized coordinates are used, the energy can be written in a general form asE =
H(Q,P,t).Equation (3.11) showed that even though bothQandPdepend on time, the
energy depends on time only ifHexplicitly depends on time. Thus, in a situation whereH
does not explicitly depend on time, the energy would have the formE=H(Q(t),P(t)) =
const.
It is important to realize that both canonical momentum and energy depend on the
reference frame. For example, a bullet fired in an airplane in the direction opposite to
the airplane motion and with a speed equal to the airplane’s speed, hasa large energy as
measured in the airplane frame, but zero energy as measured by an observer on the ground.
A more subtle example (of importance to later analysis of waves and Landaudamping)
occurs whenAand/orφhave a wave-like dependence, e.g.φ(x,t) =g(x−Vpht)where
Vphis the wave phase velocity. This potential istime-dependentin the lab frame and
so the associated Lagrangian has an explicit dependence on time in the lab frame, which
implies thatenergy is not a constant of the motion in the lab frame. In contrast,φistime-
independentin the wave frame and so the energy is aconstant of the motion in the wave
frame. Existence of a constant of the motion reduces the complexity of the system of
equations and typically makes it possible to integrate at least one equation in closed form.
Thus it is advantageous to analyze the system in the frame having the most constants of the
motion.