Mathematical Principles of Theoretical Physics

3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 153

We have the following theorems for divA-free constraint variations.

Theorem 3.26.Let F:Wm,^2 (M,g)→R^1 be a functional of Riemannian metrics. Then there
is a vector fieldΦ∈H^1 (TM)such that the extremum points{gij}of F with thedivA-free
constraint satisfy the equation

(3.4.42) δF(gij) =DΦ+A⊗Φ,

where D is the covariant derivative operator as in (3.4.41).

Theorem 3.27.Let F: Hm(TM)→R^1 be a functional of vector fields. Then there is a
scalar functionφ∈H^1 (M)such that for a given vector field A, the extremum points u of F
with thedivA-free constraint satisfy the equation

(3.4.43) δF(u) = (∂+A)φ.

Proof of Theorems3.26and3.27.First we prove Theorem3.26. By (3.4.40), the extremum
points{gij}ofFwith the divA-free constraint satisfy
∫

M

δF(gij)·X

√

−gdx= 0 ∀X∈L^2 (T 02 M)with divAX= 0.

It implies that

(3.4.44) δF(gij)⊥L^2 D(T 20 M) ={v∈L^2 (T 20 M)|divAv= 0 }.

By Theorem3.17,L^2 (T 20 M)can be orthogonally decomposed into

L^2 (T 20 M) =LD^2 (T 20 M)⊕G^2 (T 20 M),

G^2 (T 20 M) ={DAΦ|Φ∈H^1 (T 10 M)}.

Hence it follows from (3.4.44) that

δF(gij)∈G^2 (T 20 M),

which means that the equality (3.4.42) holds true.
To prove Theorem3.27, for an extremum vector fielduofFwith the divA-free constraint,
we derive in the same fashion thatusatisfies the following equation

(3.4.45)

∫

M

δF(u)·X

√

−gdx= 0 ∀X∈L^2 (TM)with divAx= 0.

In addition, Theorem3.17means that

L^2 (TM) =L^2 D(TM)⊕G^2 (TM),

L^2 D(TM) ={v∈L^2 (TM)|divAv= 0 },

G^2 (TM) ={DAφ|φ∈H^1 (M)}.

Then we infer from (3.4.45) that

δF(u)∈G^2 (TM).

Thus we deduce the equality (3.4.43).
The proofs of Theorems3.26and3.27are complete.