Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 145


whereδFis the derivative operator ofF.
Given a variational problem, it is important to computeδFfor a given functionalF.
Hereafter we give a brief introduction of general methods tocompute the derivative operators
fromF.
LetFbe the functional given by (3.4.1), andX∗be the dual space ofX. The derivative
operatorδF(u)ofFatu∈Xis a linear functional onX, i.e.δF(u)∈X∗for eachu∈X. In
other words,δFis a mapping fromXtoX∗:


(3.4.4) δF:X→X∗.


Denote〈·,·〉the product betweenXandX∗, i.e.


〈·,·〉:X×X∗→R.

Then the derivative operatorδFin (3.4.4) satisfies the relation:


(3.4.5) 〈δF(u),v〉=


d




λ= 0

F(u+λv) foru,v∈X,

whereλ∈R^1 is real number.
Based on (3.4.5), it is easy to see that the minimal pointusatisfying (3.4.2) is a solution
of the variational equation (3.4.3). In fact, by (3.4.2) for any givenv∈Xthe functionf(λ) =
F(u+λv)is minimal atλ=0:


d f( 0 )

= 0 ⇒


d




λ= 0

F(u+λv) = 0 ∀v∈X.

It follows then by (3.4.5) that


〈δF(u),v〉= 0 ∀v∈X,

which means thatusatisfies (3.4.3).
In the following, we give a simple example to show how to computeδFusing formula
(3.4.5).
LetX=H^1 (Rn), andF:H^1 (Rn)→R^1 be given by


(3.4.6) F(u) =



Rn

[


1


2


|∇u|^2 +f(u)

]


dx foru∈H^1 (Rn).

We see that


d

F(u+λv) =

d


Rn

[


1


2


|∇u+λ∇v|^2 +f(u+λv)

]


dx

=



Rn

[


(∇u+λ∇v)·∇v+f′(u+λv)v

]


dx.

Hence we have


d




λ= 0

F(u+λv) =


Rn

[


∇u·∇v+f′(u)v

]


dx

=(by the Gauss formulas( 3. 2. 36 ))

=


Rn

[


−div(∇u)+f′(u)

]


vdx.
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