Beams and Beam Physics
3
Lorentz force law can also be obtained from the Lagrangian
L = −mc
2
1 −
v 2
c 2 + qq v ·
A − qV ;
(1.2)
refer to eqs. (1.85), (1.145) in [5], and, for example, [29]. From this Lagrangian, one can also obtain a Hamiltonian of the motion in a procedure
that is standard for all Lagrangian systems. One begins by defining the canonical momentum as:
p can =
∂L
∂∂ v
,
which here has the form
p can = γmm v + q
A = p dyn + q
A,
(1.3)
where
γ =
1
1 − v 2 /c 2
,
and the canonical momentum
p can is different from the relativistic dynamical
momentum
p dyn = γmm v.
(1.4)
The Hamiltonian of the motion can then be found as
H = p can · v − L.
This expression initially contains both
p can and v, and it is necessary to eliminate v and express it in terms of p can . Because p dyn = γmm v and
p dyn =
p can − q
A from eq. (1.3), γmm v = mm v/
1 − v 2 /c 2 =
p can − q
A, leading to
m
2 v
2 /(1 − v
2 /c
2 ) = ( p can − q
A)
2 , so we find
v = c ·
p can − q
A
p can − q
A
2
+ m 2 c 2
= c ·
p dyn
p 2
dyn + m 2 c 2
,
(1.5)
where the expression in terms of p dyn is listed as well. Using this,
1 − v 2 /c 2
in the Lagrangian L in eq. (1.2) is expressed in terms of
p can , also in terms
of p dyn , as
1
γ
=
1 −
v 2
c 2 =
mc
p can − q
A
2
+ m 2 c 2
=
mc
p 2
dyn + m 2 c 2
.
(1.6)
Thus, we obtain for the Hamiltonian
H = c ·
p can − q
A
2
+ m 2 c 2 + qV ;