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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