Self-Consistent Field Theory and Its Applications by M. W. Matsen

76 1 Self-consistent field theory and its applications
derivative, f ′ (x). To calculate its functional derivative, we start with
I[f + ɛδ]
≡
≈
∫ b
a
dxL(x, f(x)+ɛδ(x − y),f ′ (x)+ɛδ ′ (x − y))
I[f]+ɛ
ɛ
∫ b
a
∫ b
a
dx δ(x − y) ∂
∂f L(x, f(x),f′ (x)) +
dx δ ′ (x − y) ∂
∂f ′ L(x, f(x),f′ (x))
= I[f]+ɛ ∂
∂f L(y, f(y),f′ (y)) −
ɛ d ( )
∂
dy ∂f ′ L(y, f(y),f′ (y))
(1.328)
In the last step, the integration involving δ ′ (x − y) is converted to one involving δ(x − y)
by performing an integration by parts, and then both integrals are evaluated using the sifting
property of the Dirac delta function,
∫
dx δ(x − y)g(x) =g(y) (1.329)
where g(x) is an arbitrary function. Inserting Eq. (1.328) into the definition of a functional
derivative, we have
DI[f]
Df(x) = ∂
∂f L(x, f(x),f′ (x)) − d ( )
∂
dx ∂f ′ L(x, f(x),f′ (x))
(1.330)
This more general result is indeed consistent with our prior example. If we choose L(x, f, f ′ )=
exp(f), then ∂
∂f L(x, f, f ′ )=exp(f) and
∂
∂f
L(x, f, f ′ )=0. Substituting these into Eq.
′
(1.330) reduces it to the specific case of Eq. (1.326).
Many problems in physics can be expressed in terms of either a minimization or maximization
of an integral of the form in Eq. (1.327). One such example involves the Lagrangian
formalism of classical mechanics (Goldstein 1980), where the trajectory of an object, r p (t),
is determined by minimizing the so-called action, S, which is an integral over time involving
its position, r p (t), and velocity, r ′ p(t). The condition for an extremum of the integral, I[f], is
simply
( )
d ∂
dx ∂f ′ L(x, f(x),f′ (x)) − ∂
∂f L(x, f(x),f′ (x)) = 0 (1.331)
which is a differential equation referred to as the Euler-Lagrange equation. This result is used
numerous times throughout the Chapter.
Of course, calculus involves integration as well as differentiation. Here we demonstrate
some of the intricacies of functional integration by developing a useful expression for the
Dirac delta functional, δ[f]. This is a generalization of the ordinary delta functional defined
in Eq. (1.324), although we now use the alternative integral representation,
δ(x) = 1 ∫ ∞
dk exp(ikx) (1.332)
2π
−∞