MAXWELL-LORENTZ EQUATIONS IN GENERAL
FRENET-SERRET COORDINATES
Andreas C. Kabel
∗
Stanford Linear Accelerator Center, Stanford, CA, USA
andreas.kabel@slac.stanfor d.ed u
Abstract
We consider the trajectory of a charged particle in an
arbitrary external magnetic field. A local orthogonal co-
ordinate system is given by the tangential, curvature, and
torsion vectors. We write down Maxwell’s equations in
this coordinate system. The resulting partial differential
equations for the magnetic fields fix conditions among its
local multipole components, which can be viewed as agen-
eralization of the usual multipol expansion of the fields of
magnetic elements.
MOTIVATION
The problem at hand came about while implementing a
generalized co-moving coordinate system in TraFiC4 [1],
which, in the generalized case, is curvilinear and non-
orthogonal.
The usual approach to describing the field content of
magnetic elements in accelerator physics is the expansion
in multipoles. This is based on the fact that the magnetic
field in vacuum can be derived from a potential obeying
Laplace’s equation; assuming a symmetry along one axis,
it reduces to the two-dimensional Laplace equation, which
is solved by analytic functions of a complex variable.
Strictly speaking, this approach is only admissible in
straight section, where the co-moving coordinate system
is cartesian. In curved sections, one has to use curvilinear
coordinates, and the Laplace operator changes its shape,
leading to a different set of solutions.
Furthermore, the transformation to curvilinear changes
the equation of motion, introducing inertial forces.
In this paper, we write down the Maxwell-Lorentz equa-
tions for the case of an external purely magnetic field in a
coordinate system co-moving on the orbit induced by the
external field. We describe that orbit in terms of its local
curvature and torsion; the Laplace and Lorentz equations
are given to all orders in this frame, Laplace’s equation is
solved for two special cases.
FRENET-SERRET COORDINATES
We consider the orbit particle in the usual description
of an accelerator. Ignoring energy changes, its trajectory
is completely determined by its initial conditions and the
external magnetic field.
We use the arc length s to parametrize its trajectoryr(s).
Then, we define the usual local dreibein by the Frenet field
∗
Work supported by Department of Energy contract
76SF00515.
frame:
E
s
:=
t(s)=r
(s)
E
x
:= n(s)=
1
k
t
(s)
E
y
:=
b(s)=
t(s) × n(s)
where k is the curvature. We also introduce the torsion
w =
1
k
2
det(r
,r
,r
)
We express all quantities in the coordinate system
spanned by
E
i
, i.e. a vector
R is decomposed as
R(s, x, y)=r(s)+xn(s)+y
b(s).
The magnetic field is scaled such that
t
=
t ×
B
i. e., we absorb charge and energy into the field.
After some algebra, we find
¯
B
y
= −k
¯
B
s
= −w
¯
B
x
=0
(1)
where the index labels components with respect to
E
i
and
barred quantities are values on the trajectory.
THE EQUATIONS OF MOTION
We now look at a particle with trajectory x(t)=
x(t)n(s(t))+y(t)
b(s(t))+r(s(t)), where the coordinate s
is implicitly defined by
t(s(t)) · (r(s(t)) − x(t)) = 0. The
equations of motion for the coordinates x, y, s read:
˙x = v
x
+ yw ˙s
˙y = v
y
− xw ˙s
˙s =
v
s
1 − kx
and
¨x = a
x
+ x(˙s
2
(w
2
+ k
2
)) + 2 ˙yw ˙s+ yw¨s +(yw
− k)˙s
2
¨y = a
y
− 2˙xw ˙s + yw
2
˙s
2
− xw
˙s
2
− xw¨s
¨s =
a
s
+2˙xk ˙s +(xk
− ywk)˙s
2
1 − kx
where v
i
and a
i
are the components of the particle’s veloc-
ity and external acceleration.
DE–AC02–
Stanford Linear Accelerator Center, Stanford University, Stanford, California, 94309
SLAC-PUB-10746To appear in the Proceedings of Particle Accelerator Conference (PAC 03), Portland, OR, 12-16 May 2003.
1
Putting in the Lorentz equation with magnetic fields B
i
,
we have
a
x
=(˙y + xw ˙s)vB
s
− ˙s(1 − kx)vB
y
a
y
=(1− kx)vB
x
− (˙x − yw˙s)vB
s
a
s
=(˙x − yw˙s)vB
y
− (˙y + xw ˙s)vB
x
where v = v
Orbit
s
. The expressions for the second deriva-
tives of the coordinates contain centrifugal (proportional to
x, y) and Coriolis (proportional to ˙x, ˙y, ˙s) and higher-order
terms. Theyreduced to small quantities (O(v−v
s
),O(kx))
by the Lorentz force for the values given in (1).
THE OFF-ORBIT MAGNETIC FIELD
We are now interested in the external components of the
acceleration. They are given by the external magnetic field;
they can, however,not be arbitrary as the magneticfield has
to fulfill Maxwell’s equations. To find the conditions for
b
x
,b
y
,b
s
, we write down the metric tensor for the curvilin-
ear coordinate system defined above:
g
ik
=
h
2
+ w
2
(x
2
+ y
2
) −wy wx
−wy 10
wx 01
ik
(where x
1
= s, x
2
= x, x
3
= y and h =1− k(s)x) with
det g =(1− kx)
2
The inverse is
g
ik
= h
−2
1 wy −wx
wy h
2
+ w
2
y
2
−w
2
xy
−wx −w
2
xy h
2
+ w
2
x
2
ik
The coordinate region we are interested is free of cur-
rents, so the magnetic field is the gradient of a potential Φ
with ∆Φ = 0. We express Laplace’s operator in curvilin-
ear coordinates; we find the following Laplace operators
for the cases of vanishing torsion and constant curvature;
constant curvature and torsion; and arbitrary curvature and
torsion, resp.:
∆
k,0
= h
−2
∂
2
s
+ ∂
2
x
+ ∂
2
y
− k/h∂
x
∆
k(s),0
=∆
k,0
+ xh
−3
k
(s)∂
s
∆
k,w
=∆
k,0
+ wh
−3
(2[ky + h(y∂
x
− x∂
y
)]∂
s
+
w[−y∂
y
+ h(x
2
∂
2
x
− 2xy∂
x
∂
y
+ y
2
∂
2
y
)]
∆
k(s),w (s)
=∆
k,w(s)
+ xh
−3
k
(s)∂
s
(2)
Note that the s derivative of the torsion does not enter the
equations.
The magnetic field is
B
i
= g
ik
∂
k
Φ
so Φ has to fulfill the conditions
∂
s
Φ=−w
∂
x
Φ=0
∂
y
Φ=−k
at the origin of the local system.
Obviously, the usual analyticity property of the solutions
is lost for nonvanishing curvature.
Constant Curvature
Let us solve the constant curvature case first. We notice
that the partial differential equation separates, i. e.
Ψ(s, x, y)=ψ
s
(s)ψ
x
(x)ψ
y
(y)
We notice that ψ(s)=const because of the vanishing
torsion of the orbit. ψ
y
(y) has to be a function fulfilling
ψ
y
(y)=µψ
y
(y)
and
(1 − kx)ψ
x
(x) − µ(1 − kx)ψ
x
(x) − ψ
x
(x)=0
which is solved by Bessel functions
ψ
x
= c
1
J
0
−
µ
k
2
(1 − kx)
+c
2
Y
0
−
µ
k
2
(1 − kx)
so
Ψ(s, x, y)=
dµ(c
1,y
(µ)e
−µy
+ c
2,y
(µ)e
µy
)
c
1,x
(µ)J
0
−
µ
k
2
(1 − kx)
+c
2,x
(µ)Y
0
−
µ
k
2
(1 − kx)
This, of course, is just the well-known radial solution of
Laplace’s equation in cylinder coordinates [2].
Constant Curvature and Torsion
This case corresponds to an infinitely extending helical
orbit. The differential equation is given by line 2 of (2).
We make a separation Ansatz ψ(s, x, y)=−ws+ ϕ(x, y).
The resulting differential equation for ϕ reads
∂
2
x
+ ∂
2
y
− k/h∂
x
+
w
2
h
−3
(2[h(x∂
y
− y∂
x
) − ky]+
h(x
2
∂
2
x
− 2xy∂
x
∂
y
+ y
2
∂
2
y
)
ϕ(x, y)=0
This equation is not solvable in closed form (the coordi-
nate system is not one of the ones in which the Laplace
equation is known to be separable[2]), we can, however,
obtain a recursion relation for coefficients in a power series
2
ϕ(x, y)=
m,n
a
m,n
x
m
y
n
. After some tedious work,
one obtains
a
m−3,n+2
k(w
2
+ k
2
)(n +1)(n +2)+
− a
m−2,2+n
(w
2
+3k
2
)(n +1)(n +2)+
− a
m−1,n
k
−k
2
(m − 1)
2
+ w
2
(m − 1)(2n +1)
+
3 a
m−1,2+n
k
n
2
+3n +2
+
a
m,n
k
2
(−3m
2
+2m)+w
2
(m + n +2nm)
+
− a
m,2+n
(n +2)(n +1)+
a
1+m,−2+n
kw
2
m
2
− 1
+
a
1+m,n
k(3m +1)(m +1)+
− a
2+m,−2+n
w
2
(m +1)(m +2)+
− a
2+m,n
(m +1)(m +2)=−2w
2
kδ
m,0
δ
n,1
(3)
where the rhs term comes from the separation Ansatz. The
inital conditions are a
0,0
=0(a global gauge fixing),
a
1,0
=0, a
0,1
= −k. Note that there are special cases
of the recursion relation for m ≤ 3 or n ≤ 3, they are
(using the initial conditions)
a
2,0
+ a
0,2
=
ka
1,1
− 2a
2,1
− 6a
0,3
+ w
2
k =
− 2a
1,2
+8ka
2,0
− 6a
3,0
+6ka
0,2
=
4w
2
a
1,1
+18ka
0,3
+8ka
2,1
− 6a
3,1
− 2k
2
a
1,1
− 6a
1,3
=
− 6k
2
a
0,2
+6ka
1,2
− 12a
4,0
− 2w
2
a
0,2
−
10k
2
a
2,0
+2w
2
a
2,0
− 2a
2,2
+21ka
3,0
=
− 3kw
2
a
1,1
− 18k
2
a
0,3
+21ka
3,1
− 6w
2
a
0,3
−
12a
4,1
+7w
2
a
2,1
−10k
2
a
2,1
−6a
2,3
+k
3
a
1,1
+18ka
1,3
=0
and, for p>0
w
2
pa
0,1+p
− kw
2
a
1,−1+p
+ ka
1,1+p
+ w
2
a
0,1+p
−
6a
0,3+p
−5a
0,3+p
p−a
0,3+p
p
2
−2w
2
a
2,−1+p
−2a
2,1+p
=0
3w
2
pa
1,1+p
+18ka
0,3+p
+15ka
0,3+p
p +3ka
0,3+p
p
2
+
8ka
2,1+p
− 6a
1,3+p
− 5a
1,3+p
p − a
1,3+p
p
2
−
6a
3,1+p
+4w
2
a
1,1+p
− 6w
2
a
3,−1+p
− 2k
2
a
1,1+p
=0
− 2kw
2
pa
1,1+p
− 5w
2
a
0,3+p
p − w
2
a
0,3+p
p
2
+
3ka
1,3+p
p
2
+15ka
1,3+p
p − 15k
2
a
0,3+p
p − 3k
2
a
0,3+p
p
2
+
5w
2
pa
2,1+p
− 3kw
2
a
1,1+p
+3kw
2
a
3,−1+p
− a
2,3+p
p
2
−
5a
2,3+p
p − 6w
2
a
0,3+p
− 10k
2
a
2,1+p
+21ka
3,1+p
+
18ka
1,3+p
+7w
2
a
2,1+p
+ k
3
a
1,1+p
−
18k
2
a
0,3+p
− 6a
2,3+p
− 12w
2
a
4,−1+p
− 12a
4,1+p
=0
and
− kw
2
pa
1+p,0
+ k
3
a
1+p,0
p
2
+ w
2
a
2+p,0
p+
16ka
3+p,0
p +3ka
3+p,0
p
2
− kw
2
a
1+p,0
+2k
3
pa
1+p,0
+
2kw
2
a
−1+p,2
− 3p
2
k
2
a
2+p,0
− a
4+p,0
p
2
− 7a
4+p,0
p−
2w
2
a
p,2
+6ka
1+p,2
− 10k
2
a
2+p,0
− 6k
2
a
p,2
− 12a
4+p,0
+
2w
2
a
2+p,0
+ k
3
a
1+p,0
+2k
3
a
−1+p,2
+21ka
3+p,0
−
11k
2
pa
2+p,0
− 2a
2+p,2
=0
−3kw
2
pa
1+p,1
+16ka
3+p,1
p−3kw
2
a
1+p,1
+2k
3
pa
1+p,1
−
11k
2
pa
2+p,1
−3p
2
k
2
a
2+p,1
+6kw
2
a
−1+p,3
+k
3
a
1+p,1
p
2
+
3w
2
a
2+p,1
p +3ka
3+p,1
p
2
− a
4+p,1
p
2
− 7a
4+p,1
p+
18ka
1+p,3
− 6w
2
a
p,3
+6k
3
a
−1+p,3
− 10k
2
a
2+p,1
+
k
3
a
1+p,1
− 12a
4+p,1
− 18k
2
a
p,3
+
21ka
3+p,1
+7w
2
a
2+p,1
− 6a
2+p,3
=0
− 3kw
2
a
1,1
− 18k
2
a
0,3
+21ka
3,1
− 6w
2
a
0,3
− 12a
4,1
+
7w
2
a
2,1
− 10k
2
a
2,1
− 6a
2,3
+
k
3
a
1,1
+18ka
1,3
=0
(3) can be viewed as a generalization of the multipole
Ansatz for helical orbits. Of course, setting w = k =0
we obtain the recursion relation valid for harmonic func-
tions in x, y, namely (n +1)(n+2)a
m,2+n
+(m+1)(m +
2)a
2+m,n
=0
. Obviously, the complexity of the power series prescrip-
tion limits its usefulness.
Planar Undulator
Here, we have vanishingtorsion and a periodic curvature
k(s)=κ exp(−iλs). We substitute k(s) for s and obtain,
after separating off the harmonic y dependence as ∂
2
y
ψ =
−µψ, where µ =0due to the boundary condition w =0
(−λ
2
k(∂
k
+ hk∂
2
k
)+h
2
∂
x
h∂
x
)ψ(k, x)=0 (4)
Again, a solution is only possible in terms of a power se-
ries in terms of x and k, similiar as above. We obtain the
recursion relation
a
m−1,n−3
(m − 1)
2
− a
m−1,n−1
λ
2
(n − 1)(n − 2)−
a
m,n−2
m(3m − 1) + λ
2
a
m,n
n
2
+
a
m+1,n−1
(3m +1)(m +1)−
a
m+2,n
(m +1)(m +2)=0 (5)
, as well as special cases for small m, n and for initial con-
ditions, which will not be given here for lack of space.
REFERENCES
[1] A. Kabel. Particle tracking and bunch population in trafic4
2.0. 2003. this conference.
[2] Philip McCord Morse and Herman Feshbach. Methods of
theoretical physics. 1953.
3