 CHAPTER 3
Solving for the Scalar
Magnetic Potential
3.1 THE "VARIATIONAL" FORMULATION
We now treat the problem we arrived at for what it is, an
equation,
to be
studied and solved as such: Given a bounded domain D, a number I
(the mmf), and a function t-t (the permeability), subject to the conditions
0 < l.t 0 < ~ < bt 1 of Eq. (18) in Chapter 2,
(1)
find ~
~
(I)I=
{(4) E
(I}" (4)=
0 on sh0, q)= I
on
Shl }
such that
fD ~ grad q0. grad q0' = 0 V q0' e ~0.
S b
FIGURE 3.1. The situation, reduced to its meaningful geometrical elements.
All potentials q0 and test functions q0' belong to the encompassing
linear space of piecewise smooth functions on D (cf. 2.4.2), and the
geometrical elements of this formulation, surface S = S h u S b partition
h h h . . h
S = S o u S 1 of the magnetic wall S (Fig. 3.1), are all that we abstract
from the concrete situation we had at the beginning of Chapter 2. We
note that the magnetic energy (or rather, coenergy, cf. Remark 2.6) of h =
grad q0, that is,
61 62 CHAPTER 3 Solving for the Scalar Magnetic Potential
1 fD~
Igrad
I 2,
F( 0) =
is finite for all elements of cI). The function F, the type of which is
FIELD ~ REAL,
and more precisely, (I) --~ IR, is called the
(co)energy
functional.
Remark 3.1. The use of the quaint term "functional" (due to Hadamard),
not as an adjective here but as a somewhat redundant synonym for
"function", serves as a reminder that the argument of F is not a simple
real- or vector-valued variable, but a point in a space of infinite dimension,
representative of a field. This is part of the "functional" point of view
advocated here: One
may
treat complex objects like fields as mere "points"
in a properly defined functional space. 0
Function F is quadratic with respect to q0, so this is an analogue, in
infinite dimension, of what is called a
quadratic form
in linear algebra.
Quadratic forms have associated polar forms. Here, by analogy, we
define the
polar form
of F as Y-(q0, ~) = ~D ~t grad q). grad ~¢, a bilinear
function of two arguments, that reduces to F, up to a factor 2, when both
arguments take the same value.
The left-hand side of (1) is thus y- (q0, q0'). This cannot be devoid of
significance, and will show us the way: In spite of the dimension being
infinite, let us try to apply to the problem at hand the body of knowledge
about quadratic forms. There is in particular the following trick, in which
only the linearity properties are used, not the particular way F was
defined: For any real 5~,
(2) 0 < F(q0 + k~) : F(q0) + 5~ y- (q0, ~)
+ ~2
F(~) V ~ ~ ~.
One may derive from this, for instance, the Cauchy-Schwarz inequality,
by noticing that the discriminant of this binomial function of ~ must be
nonpositive, and hence
y- (q0, ~) < 2 [F(q))] 1/2 [F(I]/)] 1/2,
with equality only if ~1/= aq0 + b, with a and b real, a > 0. Here we
shall use (2) for a slightly different purpose:
Proposition 3.1.
Problem
(1)
is equivalent to
(3)
Find (p
~ (D I
such that
F(q))< F(~) V ~ ~ cI) ~,
the
coenergy minimization
problem.
Proof.
Look again at Fig. 2.8, and at Fig. 3.2 below. If q0 solves (3), then
F(q0) < F(tp + ;~q0') for all q0' in cI) °, hence 5~ 7 (q0, q0') + 5~2 F(q0') > 0 for all

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