Theoretical and Mathematical Physics, 185(2): 1557–1581 (2015)
CONSERVATION LAWS, DIFFERENTIAL IDENTITIES, AND CONSTRAINTS OF PARTIAL DIFFERENTIAL EQUATIONS V. V. Zharinov∗
We consider specific cohomological properties such as low-dimensional conservation laws and differential identities of systems of partial differential equations (PDEs). We show that such properties are inherent to complex systems such as evolution systems with constraints. The mathematical tools used here are the algebraic analysis of PDEs and cohomologies over differential algebras and modules.
Keywords: differential algebra, conservation law, differential identity, differential constraint
1. Introduction This essentially review paper is based on our papers published during the last thirty years [1]–[7]. It treats specific properties of systems of partial differential equations (PDEs) such as low-dimensional conservation laws and differential identities. It seems that such properties are inherent to nonevolutiontype systems. We say that a system of PDEs is of the evolution type if it can be written as an evolution system after a suitable change of variables and of the nonevolution type otherwise. In Sec. 2, we define all necessary algebraic ingredients (differential algebras, differential ideals, differential modules, differential polynomials, and differential complexes) and give the relevant cohomological results (Green’s formula and the exactness of some important sequences). In Sec. 3, we describe general systems of PDEs in the algebraic settings, consider the basic elements of this setting (solution manifold, differential algebra, differential ideal, differential complex, low-dimensional conservation laws, and differential identities associated with a given system of PDEs), and derive relations between the spaces of conservation laws and differential identities. In Sec. 4, we show that evolution systems do not have low-dimensional conservation laws and study evolution systems with constraints (which are of the nonevolution type). It turns out that such systems may have low-dimensional conservation laws depending on the cohomological properties of the subsystem of constraints. The main results here are Theorems 10, 11, and 12. Further, we consider systems of continuity equations and calculate their conservation laws (Theorem 13). Based on this result, we calculate conservation laws for an evolution system with a zero-divergence constraint. In particular, we show that such systems have nontrivial low-dimensional conservation laws (Theorem 14). The general results are illustrated by several instructive examples including Yang–Mills and Maxwell’s equations. ∗
Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia, e-mail:
[email protected]. This research was funded by a grant from the Russian Science Foundation (Project No. 14-50-00005).
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 2, pp. 227–251, November, 2015. Original article submitted May 05, 2015. c 2015 Pleiades Publishing, Ltd. 0040-5779/15/1852-1557
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2. Preliminaries 2.1. Notation. We use the following standard notation: • R and C are the fields of all real and complex numbers, • Z, Z+ , and N are the sets of all integers, nonnegative integers, and natural numbers (positive integers), 1 m μ • I = Zm + = {i = (i , . . . , i ) | i ∈ Z+ , 1 ≤ μ ≤ m}, m ∈ N, is the additive Abelian monoid of all multi-indices with the componentwise operations, (i1 , . . . , im ) + (j 1 , . . . , j m ) = (i1 + j 1 , . . . , im + j m ),
• special multi-indices are 0 = (0, . . . , 0) and (μ) = (0, . . . , 0, 1, 0, . . . , 0), where 1 is the μth entry, 1 ≤ μ ≤ m, and • |i| = i1 + · · · + im for any i = (i1 , . . . , im ) ∈ I. Further, • X = Rm = {x = (x1 , . . . , xm ) | xμ ∈ R, 1 ≤ μ ≤ m}, m ∈ N, is the space of independent variables, • U = RA = {u = (uα ) | uα ∈ R, α ∈ A} is the space of dependent variables, where A is a set of indices, α α • U = RA I = {u = (ui ) | ui ∈ R, α ∈ A, i ∈ I} is the space of differential variables, the projection α α α π : U → U, u = (uα i ) → u = (u ), where u = u0 ,
• XU = X × U is the space of all variables, • ∂xμ , ∂uαi are the partial derivatives, 1 ≤ μ ≤ m, α ∈ A, i ∈ I, • Dμ = ∂xμ +
α∈A, i∈I
α uα i+(μ) ∂ui are the total partial derivatives, 1 ≤ μ ≤ m, and
ˇμ · · · ∧ dxm (here and hereafter, a “checked” • dm x = dx1 ∧ · · · ∧ dxm , dμ x = (−1)μ−1 dx1 ∧ · · · dx argument is understood to be omitted). In addition, • Φ = C ∞ (X; R) is the algebra of all smooth functions from X to R, • ΦA = C ∞ (X; U) is the Φ-module of all smooth functions from X to U, ∞ • ΦA I = C (X; U) is the Φ-module of all smooth functions from X to U, ∞ • F = Cfin (XU; R) is the algebra of all smooth functions from XU to R depending on a finite number of arguments (see, e.g., [3], [8] for more details), ∞ • F K = Cfin (XU; RK ) is the F -module of all smooth functions from XU to RK depending on a finite number of arguments, where K is a set of indices, and ∞ K • FIK = Cfin (XU; RK I ) is the F -module of all smooth functions from XU to RI depending on a finite number of arguments.
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2.2. Conventions. • All constructions here are local without this being explicitly noted. For example, we write ϕ ∈ Φ assuming that in fact ϕ ∈ Φ(O) = C ∞ (O; R), where O is a domain in X. • All linear operations are performed over the field R of real numbers, although this is inessential. • As a rule, we assume summation over repeating upper and lower indices in the natural limits. • Braces { · } denote symmetrization, and square brackets [ · ] denote alternation. • Angle brackets · , · denote the pairing of elements from dual modules. • The symbol a denotes composition of maps. • We assume that the differential algebra (A, ∂) under study is nondegenerate (see Sec. 2.6 for more details). 2.3. Differential algebras. A differential algebra (see, e.g., [9], [10]) is a pair (A, ∂), where • A is an associative commutative unital algebra and • ∂ = (∂1 , . . . , ∂m ) is an A-linearly independent subset of D(A) such that [∂μ , ∂ν ] = 0, 1 ≤ μ, ν ≤ m. We recall that for any associative commutative unital algebra A, the set D(A) of all its derivations is defined, i.e., the set of all endomorphisms X ∈ EndR (A) such that the Leibniz rule X(ab) = (Xa) b + a (Xb) holds for all a, b ∈ A. The set D(A) has the natural structure of a Lie algebra over the algebra A, i.e., it has two consistent structures: the structure of a Lie algebra and the structure of an A-module with the consistency relation [aX, Y ] = a[X, Y ] − (Y a)X for all a ∈ A and X, Y ∈ D(A). Let • d = {ζ = ζ μ ∂μ | ζ μ ∈ A} be the subalgebra of the Lie algebra D(A) with the A-basis ∂ = (∂1 , . . . , ∂m ) and • d∗ = HomA (d; A) = {ω = ωμ θμ | ωμ ∈ A} be the dual A-module with the dual A-basis θ = (θ1 , . . . , θm ), where θμ , ∂ν = θμ (∂ν ) = δνμ for all 1 ≤ μ, ν ≤ m. 2.4. Differential ideals. Let (A, ∂) be a differential algebra. An ideal I of the algebra A is called a differential ideal (or (A, ∂)-ideal) if it is closed under the action of derivations ∂1 , . . . , ∂m , i.e., if ∂(I) ⊂ I. Clearly, the pair (I, ∂), where ∂ = ∂ I , is a differential algebra. Let I be an (A, ∂)-ideal. The quotient differential algebra (A, ∂) is defined by the exact sequence of A-modules 0 → I → A → A → 0, a → a = a + I, and by the quotient rule ∂ μ a = ∂μ a for all a ∈ A, 1 ≤ μ ≤ m. 2.5. Differential modules. Let (A, ∂) be a differential algebra. An A-module M is called a differential module (or (A, ∂)-module) if the action d → gl(M) is defined such that the Leibniz rule ∂μ (aM ) = (∂μ a)M + a(∂μ M ) holds for all 1 ≤ μ ≤ m, a ∈ A, and M ∈ M. We recall that gl(M) is the Lie algebra adjoint to the associative algebra EndR (M) of all linear endomorphisms of M. A submodule S of an (A, ∂)-module M is called a differential submodule (or (A, ∂)-submodule) if it is closed under the action of derivations ∂1 , . . . , ∂m , i.e., if ∂(S) ⊂ S. In this case, the quotient differential module M is defined by the exact sequence of A-modules 0 → S → M → M → 0,
M → M = M + S,
and by the quotient rule ∂ μ M = ∂μ M for all M ∈ A, 1 ≤ μ ≤ m. 1559
2.6. Differential polynomials. Let (A, ∂) be a differential algebra. Let A[∂] denote the set of all differential polynomials, i.e., the A-module of all polynomials in the indeterminates ∂1 , . . . , ∂m with coefficients in A. Every polynomial P = P (∂) ∈ A[∂] can be written in two ways:
P (∂) =
ai ∂i =
|i|≤ord(P )
aμ1 ...μp ∂μ1 ...μp ,
0≤p≤ord(P )
where • the number ord(P ) ∈ Z+ is the order of P (∂), • the coefficients ai , aμ1 ...μp ∈ A, i ∈ I, 1 ≤ μ1 , . . . , μp ≤ m, • aπ(μ1 ...μp ) = aμ1 ...μp for any permutation π ∈ Σp , 1
m
• a monomial ∂i = (∂1 )i · · · (∂m )i , i = (i1 , . . . , im ) ∈ I, and • a monomial ∂μ1 ...μp = ∂μ1 · · · ∂μp , 1 ≤ μ1 , . . . , μp ≤ m. The set A[∂] has the structure of an associative unital algebra. The binary operation here is specified by the Leibniz rule ∂μ a a = ∂μ a + a∂μ for all 1 ≤ μ ≤ m, a ∈ A. For example,
(a∂μ ) a (b∂ν ) = (a(∂μ b))∂ν + (ab)∂μν for all a, b ∈ A, 1 ≤ μ, ν ≤ m. Every differential polynomial P (∂) = i ai ∂i ∈ A[∂] defines a linear differential operator on A by the rule i P (∂)b = i a (∂i b), b ∈ A. We note that the bilinear operation in the algebra A[∂] coincides with the composition rule in EndR (A). The differential algebra (A, ∂) is said to be nondegenerate if for any (A, ∂)-ideal I and any differential polynomial P (∂) = i ai ∂i ∈ A[∂], the inclusion P (∂)φ ∈ I for all φ ∈ A holds iff all coefficients ai ∈ I. In this case, there is an inclusion A[∂] ⊂ EndR (A). We assume that the differential algebra under study is nondegenerate. In the same way, for every (A, ∂)-module M, the set M[∂] of all differential polynomials with coefficients in M is defined. Here, the Leibniz rule defines the structure of an A[∂]-bimodule on the set M[∂]. 2.7. Integration by parts (Green’s formula). Let (A, ∂) be a differential algebra and K be an (A, ∂)-module. The dual A-module K∗ = HomA (K; A) has the structure of an (A, ∂)-module defined by the Leibniz rule ∂μ φ, f = ∂μ φ, f + φ, ∂μ f for all 1 ≤ μ ≤ m, φ ∈ K∗ , f ∈ K. Further, let K and L be (A, ∂)-modules and K∗ and L∗ be their duals. The A-modules M = HomA (K; L) and N = HomA (L∗ ; K∗ ) have the structure of an (A, ∂)-module defined by the Leibniz rule. For example, ∂μ a M = ∂μ M + M a ∂μ
for all 1 ≤ μ ≤ m,
M ∈ M.
Moreover, there is a conjugation map M → N , M → N = M ∗ , where M ∗ φ, f = φ, M f for all φ ∈ L∗ , f ∈ K. We use the matrix notation φM = M ∗ φ. In this case, the preceding definition becomes φM, f = φ, M f . 1560
The Lagrange conjugate polynomial P ∗ (∂) =
(−1)p ∂μ1 ...μp a (M μ1 ...μp )∗ ∈ N [∂],
p
i.e., P ∗ (∂)φ =
(−1)p ∂μ1 ...μp (φM μ1 ...μp )
for all φ ∈ L∗ ,
p
is defined for every polynomial P (∂) =
p
M μ1 ...μp ∂μ1 ...μp ∈ M[∂].
Theorem 1. For any linear operator P (∂) =
M μ1 ...μp ∂μ1 ...μp ∈ HomR (K; L),
p
there exists a bilinear operator J(∂) = (J μ (∂)) ∈ HomR (L∗ × K; Am ) such that the Green’s formula φ, P (∂)f − P ∗ (∂)φ, f = ∂μ (J μ (∂)(φ, f )) holds for all pairs (φ, f ) ∈ L∗ × K. For example, we can take
J μ (∂)(φ, f ) =
(−1)p ∂μ1 ...μp (φM μμ1 ...μp ν1 ...νq ), ∂ν1 ...νq f .
p,q
Proof. The theorem can be proved by direct verification.
2.8. Differential complex. Let (A, ∂) be a differential algebra. The external algebra Ω(A) = ⊕q∈Z Ωq (A) is defined, where the A-modules are
q
⎧ ⎪ ⎪ ⎪A, ⎨
Ω (A) =
⎪ ⎪ ⎪ ⎩
q = 0,
q ∗
∧ d , 1 ≤ q ≤ m,
0,
q < 0 or q > m.
Further, the (A, ∂)-module Ω(M) = M ⊗A Ω(A) is defined for every (A, ∂)-module M. Every form ω ∈ Ωq (M), 1 ≤ q ≤ m, has the representation ω=
1 ων ...ν ⊗ θν1 ...νq , q! 1 q
ων1 ...νq ∈ M,
where • ωπ(ν1 ...νq ) = sgn(π) ων1 ...νq for all permutations π ∈ Σq , • θν1 ...νq = θν1 ∧ · · · ∧ θνq , 1 ≤ ν1 , . . . , νq ≤ m, and • (θν1 ...νq | 1 ≤ ν1 < · · · < νq ≤ m) is a basis of the A-module Ωq (A). 1561
0
0
0
0
? - Ω0 (S)
? - Ω0 (M)
? - Ω0 (M)
- 0
0
d ? - Ω1 (S)
d ? - Ω1 (M)
d ? - Ω1 (M)
- 0
d ? .. .
0
d ? .. .
d ? .. .
d d d ? ? ? - Ωm (S) - Ωm (M) - Ωm (M) ? 0
? 0
- 0
? 0
Fig. 1
The following operations are defined: • εμ : Ωq (M) → Ωq+1 (M), ω → θμ ∧ ω, • ∂μ : Ωq (M) → Ωq (M), ω → ∂μ ω, and • dq = dΩq (M) = εμ a ∂μ : Ωq (M) → Ωq+1 (M), ω → dω, where ω = (1/q!)ων1 ...νq ⊗ θν1 ...νq , ∂μ ω = (1/q!)(∂μ ων1 ...νq ) ⊗ θν1 ...νq , q ∈ Z, 1 ≤ μ ≤ m. Clearly, d a d = 0, and the differential complex {Ωq (M); dq } is hence defined with the cohomology spaces H q (M) = Ker dq / Im dq−1 . Let S be an (A, ∂)-submodule of the (A, ∂)-module M and M be the quotient (A, ∂)-module. By standard homological algebra techniques (see, e.g., [11]), the exact sequence 0 → (S, ∂) → (M, ∂) → (M, ∂) → 0 of (A, ∂)-modules defines the exact sequence q
0 → {Ωq (S); dq } → {Ωq (M); dq } → {Ωq (M); d } → 0 of complexes, which is explicitly written as the commutative diagram of linear spaces with exact rows presented in Fig. 1. Theorem 2. The sequence of linear spaces Δ
0 −→ H 0 (S) −→ H 0 (M) −→ H 0 (M) − −→ Δ
Δ
−− → H 1 (S) −→ H 1 (M) −→ H 1 (M) − −→ · · · Δ
· · · −− →H m (S) −→H m (M) −→H m (M) −→ 0 is exact, where the connecting linear maps Δ = Δq : H q (M) → H q+1 (S),
ω → Δω,
0 ≤ q ≤ m − 1,
are defined by the rule Δω = dω for all ω ∈ Ωq (M) and dω ∈ Ωq+1 (S). 1562
Proof. The theorem is proved by a standard diagram search (see [11]).
We note that H 0 (S) = {ω ∈ S | dω = 0}, H 0 (M) = {ω ∈ M | dω = 0}, and H 0 (M) = {ω ∈ M | dω ∈ S} are linear spaces of differential constants of the (A, ∂)-modules S, M, and M. 2.9. One exact sequence. Let (A, ∂) be a differential algebra and M be an (A, ∂)-module. We consider the differential complex {Ωq (M[∂]); dq }. Here, for every form ω=
1 μ1 ...μp ω ∂μ1 ...μp ⊗ θν1 ...νq ∈ Ωq (M[∂]), q! p ν1 ...νq
...μp ωνμ11...ν ∈ M, q
we have ∂μ ω =
=
dω =
1 ...μp ∂μ a ωνμ11...ν ∂μ1 ...μp ⊗ θν1 ...νq = q q! p 1
...μp ...μp ∂μμ1 ...μp ⊗ θν1 ...νq , ∂μ ωνμ11...ν ∂μ1 ...μp + ωνμ11...ν q q q! p
1 ...μp (dω)μν01...ν ∂μ1 ...μp ⊗ θν0 ...νq , q (q + 1)! p
...μp (dω)μν01...ν q
Every polynomial P (∂) =
μ1 ...μp {μ1 μ2 ...μp } = (q + 1) ∂[ν0 ων1 ...νq ] + δ[ν0 ων1 ...νq ] .
p
M μ1 ...μp ∂μ1 ...μp ∈ M[∂] has the characteristic
χ(P (∂)) =
(−1)p ∂μ1 ...μp M μ1 ...μp ∈ M.
p
Theorem 3. There exist linear maps hq : Ωq (M[∂]) → Ωq−1 (M[∂]), 0 ≤ q ≤ m, such that the homotopy formula ⎧ ⎨ω, q = m, (dq−1 a hq + hq+1 a dq )ω = ⎩ω − χ(ω), q = m, holds for every form ω ∈ Ωq (M[∂]). For example, we can take
hq ω =
⎧ ⎪ ⎨0, ⎪ ⎩
q = 0,
q μ1 ...μp 1 (h ω)ν1 ...νq−1 ∂μ1 ...μp ⊗ θν1 ...νq−1 , q > 0, (q − 1)! p
where ...μp (hq ω)μν11...ν = q−1
μλ1 ...λs μ1 ...μp (−1)s c(p, q, s)∂λ1 ...λs ωμν , 1 ...νq−1
s
c(p, q, s) =
(p + 1) · · · (s + p + 1) . (p + m − q + 1) · · · (s + p + m − q + 1)
Proof. The theorem can be proved by direct verification.
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The characteristic map χ : Ωm (M[∂]) → Ωm (M) acts by the rule ω = ω(∂) ⊗ θ∧m → χ(ω) = χ(ω(∂)) ⊗ θ∧m , where ω(∂) =
p
ω μ1 ...μp ∂μ1 ...μp ∈ M[∂] and θ∧m = θ1 ∧ · · · ∧ θm .
Corollary 1. The sequence of linear spaces 0 −→ Ω0 (M[∂])
d
d
d
χ
−→ Ω1 (M[∂]) −→ · · ·
d
· · · −→ Ωm−1 (M[∂]) −→ Ωm (M[∂]) −−→ Ωm (M) −→ 0 is exact, i.e., H q (M[∂]) = 0, q = m, H m (M[∂]) = Ωm (M[∂])/ ker χ.
3. Partial differential equations 3.1. PDEs in the algebraic approach. In the algebraic approach (see [1], [8], [12], [13]), a system of PDEs is written as F = 0, where • F = (F κ ) ∈ F K , F κ = F κ (x, u) = F κ (xμ , uα i | |i| ≤ p), κ ∈ K, • xμ are independent variables, 1 ≤ μ ≤ m, α • uα i = ∂xi u are differential variables, α ∈ A, i ∈ I, 1
m
• ∂xi = (∂x1 )i · · · (∂xm )i , and • p ∈ N is the order of the system F = 0 (see Sec. 2.1 for details of the notation). The solution manifold of the system F = 0 is the set Sol(F ) = {(x, u) ∈ XU | Di F κ (x, u) = 0, i ∈ I, κ ∈ K}, 1
m
where Di = (D1 )i · · · (Dm )i . The system F = 0 is said to be regular if the set Sol(F ) has the structure of a smooth submanifold in XU. A function φ = (φα ) ∈ ΦA is a solution of the system F = 0 if F κ (x, u)u=jφ(x) ≡ 0, i. e., {(x, u) | u = jφ(x))} ⊂ Sol(F ), A α α where the jet jφ = (φα i ) ∈ ΦI and φi = ∂xi φ . We assume that the system F = 0 is nontrivial, i.e., Sol(F ) = XU, and consistent, i.e., Sol(F ) = ∅.
3.2. The differential ideal. The pair (F , D) has the structure of a differential algebra (see Sec. 2.3), where D = (D1 , . . . , Dm ) (we note that [Dμ , Dν ] = 0 and 1 ≤ μ, ν ≤ m). Moreover, it can be verified that this algebra is nondegenerate and the subalgebra of its differential constants is H 0 (F ) = R. Every system F = 0 defines an (F , D)-ideal of the algebra (F , D), namely, the ideal I = I(F ) = {φ = P (D)F = Pκ (D)F κ | P (D) = (Pκ (D)) ∈ FK [D]}. By definition, there is the inclusion I ⊂ {φ ∈ F | φ|Sol(F ) = 0} (the inverse inclusion holds if the system is regular). The ideal I is nontrivial, i.e., I = 0, F (we recall that we assume that the system F = 0 is nontrivial). In particular, the subalgebra of its differential constants is H 0 (I) = 0. The quotient algebra F is defined by the exact sequence ι
→ F → F → 0. 0→I− The pair (F , D) has the structure of a differential algebra, where Dμ φ = Dμ φ, 1 ≤ μ ≤ m, φ ∈ F. We can regard the elements of the algebra F as smooth functions on the solution manifold Sol(F ) (this is true if the system is regular). 1564
3.3. Variational complex. In the formal variational calculus, the complex {Ωq (F ); dq } is one of the main objects and is known as the variational complex. Here, d = {ζ = ζ μ Dμ | ζ μ ∈ F },
d∗ = {ω = ων dxν | ων ∈ F },
θν = dxν (we note that dxν (Dμ ) = Dμ xν = δμν ). The following theorem is perhaps the main result in the formal variational calculus (see, e.g., [8], [12]). Theorem 4. The cohomology spaces of the complex {Ωq (F ); dq } are H 0 (F ) = R,
1 ≤ q ≤ m − 1,
H q (F ) = 0,
H m (F ) = L(F ) = F Ker E,
where the Euler operator is E : F → FA , f → Ef = (Eα f ), Eα f = δuα f =
(−1)|i| Di ∂uαi f,
α ∈ A.
i
3.4. Conservation laws and differential ideals. Let F = 0 be a system of PDEs and I be its differential ideal. For a quotient form ω ∈ Ωm−1 (F ), the condition dω = 0 means that dω ∈ Ωm (I) and hence dω Sol(F ) = 0, i.e., ω is closed on the solution manifold. In physics, such (m−1)-forms are called conserved currents, and their quotients are called conservation laws. Keeping this in mind, we can regard elements of the cohomology spaces H q (F ) as q-dimensional conservation laws. We have the usual conservation laws for q = m − 1 and low-dimensional ones for q < m − 1. Therefore, we regard the spaces CL0 (F ) = H 0 (F )/R,
CLq (F ) = H q (F ),
1 ≤ q ≤ m − 1,
as the spaces of conservation laws of the system F = 0. ι Further, the exact sequence 0 → I − → F → F → 0 yields an exact sequence of complexes (see Fig. 1), where S = I, M = F , M = F , H 0 (I) = 0, and H 0 (F ) = R. Theorem 5. The cohomology spaces of the complexes {Ωq (I); dq } and {Ωq (F ); dq } are related by Δ : CLq (F ) H q+1 (I), Δ
0 ≤ q ≤ m − 2, ι
0 → CLm−1 (F ) −→ H m (I) − → H m (F ) → H m (F ) → 0, Δ : H q (F ) → H q+1 (I),
ω → Δω = dω.
In particular, CLm−1 (F ) ker E. Proof. It suffices to use Theorems 2 and 4.
We note that here E : Ωm (F ) → FA , f dm x → Ef (cf. Theorem 4) and the quotient map is E : H m (I) → FA , f dm x → Ef (E(dΩm−1 (I)) = 0). 1565
3.5. Differential identities. Let F = 0 be a system of PDEs. By the definition of the differential ideal I(F ), the surjection F : FK [D] → I, P (D) → P (D)F , of (F , D)-modules is defined. The kernel DI = DI(F ) of this surjection is defined by the exact sequence of (F , D)-modules ι
F
0− → DI − → FK [D] − →I→ − 0. Elements of the module DI are called differential identities of the system F = 0. Again, an exact sequence of complexes (see Fig. 1) is defined, where S = DI, M = FK [D], and M = I. Theorem 6. The cohomology spaces of the complexes {Ωq (I); dq } and {Ωq (DI); dq } are related by Δ
0 → H q (I) −→ H q+1 (DI) → 0,
H 0 (DI) = 0, Δ
ι
0 ≤ q ≤ m − 2,
F
0 → H m−1 (I) −→ H m (DI) − → H m (FK [D]) − → H m (I) → 0. In particular, H m−1 (I) ker χ.
Proof. It suffices to use Theorem 1 and Corollary 1.
We note that here χ : Ωm (FK [D]) → FK , P (D)dm x → χ(P (D)) (cf. Sec. 2.9) and the quotient map is χ : H m (DI) → FK , P (D)dm x → χ(P (D)) (clearly, χ(dΩm−1 (DI)) = 0). 3.6. Conservation laws and differential identities. Let F = 0 be a system of PDEs and I be its differential ideal. Theorem 7. The relations ⎧ ⎪ H q+2 (DI), ⎪ ⎪ ⎨ CLq (F ) ker χ, ⎪ ⎪ ⎪ ⎩ ker E,
0 ≤ q ≤ m − 3, q = m − 2, q = m − 1,
hold, where χ : H m (DI) → FK and E : H m (I) → FA . Proof. It suffices to use Theorems 5 and 6.
We clarify that in the formulation of Theorem 7, • the isomorphisms CLq (F ) H q+2 (DI), 0 ≤ q ≤ m − 3, act by the rule CLq (F ) ω → dω = P (D)F → dP (D) ∈ H q+2 (DI), where ω ∈ Ωq (F ), dω = P (D)F ∈ Ωq+1 (I), P (D) ∈ Ωq+1 (FK [D]), and dP (D)F = 0, dP (D) ∈ Ωq+2 (DI); • the isomorphism CLm−2 (F ) ker χ acts by the rule CLm−2 (F ) ω → dω = P (D)F → dP (D) ∈ ker χ ⊂ H m (DI), where ω ∈ Ωm−2 (F ), dω = P (D)F ∈ Ωm−1 (I), P (D) ∈ Ωm−1 (FK [D]), dP (D)F = 0, and dP (D) ∈ ker χ = {Q(D) ∈ H m (DI) | χ(Q(D)) = 0}; and 1566
• the isomorphism CLm−1 (F ) ker E acts by the rule CLm−1 (F ) ω → dω ∈ ker E ⊂ H m (I), where ω ∈ Ωm−1 (F ), dω ∈ Ωm (I), and ker E = {φ ∈ H m (I) | Eφ = 0}. We describe the linear space ker E in more detail. We first note that Ωm (FK [D]) = FK [D] ⊗ dm x, and we therefore identify the spaces Ωm (FK [D]) and FK [D]. Now, by Green’s formula (see Sec. 2.7), every polynomial form Q(D) ∈ Ωm (FK [D]) has the representation Q(D) = χ + dJ(D), where • Q(D) = p Qμ1 ...μp Dμ1 ...μp , • χ = χ(Q(D)) = Q∗ (D)1 ∈ FK , χF = 1Q(D), F ∈ F, • J(D) = J μ (D)dμ x ∈ Ωm−1 (FK [D]), and J μ (D) = p,q (−1)p Dμ1 ...μp a Qμμ1 ...μp ν1 ...νq Dν1 ...νq . In turn, Eα (χF ) =
(−1)|i| Di ∂uαi (χκ F κ ) = i
= (−1)|i| Di (∂uαi χκ )F κ + (−1)|i| Di (∂uαi F κ )χκ = i
i
= (χ∗ F )α + (F ∗ χ)α ,
α ∈ A,
where (see [13]) χ∗ ∈ HomR (F K ; FA ) is the Lagrange conjugate to the universal linearization operator χ∗ ∈ HomR (F A ; FK ),
(χ∗ f )κ =
(∂uαi χκ )Di f α ,
f ∈ F A,
κ ∈ K,
i
and F ∗ ∈ HomR (FK ; FA ) is the Lagrange conjugate to the universal linearization operator F∗ ∈ HomR (F A ; F K ),
(F∗ φ)κ =
(∂uαi F κ )Di φα ,
φ ∈ F A,
κ ∈ K.
i
Therefore, ker E {χ ∈ FK | χ∗ F + F ∗ χ = 0}. In particular, F ∗ χSol(F ) = 0 for all χ ∈ ker E. 3.7. Examples. 3.7.1. Cauchy–Riemann system. Here, we use the complex notation • i = (i1 , . . . , im ), j = (j 1 , . . . , j m ) ∈ Zm + = I, m ≥ 2; • z = (z 1 , . . . , z m ), z = (z 1 , . . . , z m ) ∈ Cm = X, dimR X = 2m; • w, w ∈ C = U, dimR U = 2, w = (wij ), w = (w ij ) ∈ CI×I = U; • Dμ = ∂zμ + i,j (wi+(μ),j ∂wij + wi+(μ),j ∂wij ), Dμ = ∂zμ + i,j (wi,j+(μ) ∂wij + w i,j+(μ) ∂wij ), Di Dj w = wij , Di Dj w = wij , wij = w ji ; 1567
∞ • F = Cfin (XU; C), F φ = φ(z, z, w, w).
The Cauchy–Riemann system is written as CR = {∂zμ w = 0, 1 ≤ μ ≤ m} = {w0(μ) = 0, w (μ)0 = 0, 1 ≤ μ ≤ m}, i.e., F = (Fμ , F μ ) ∈ Fm , Fμ = w0,(μ) , and F μ = w(μ),0, , 1 ≤ μ ≤ m. Hence, the solution manifold is Sol(CR) = {(z, w) ∈ XU | wi,j+(μ) = 0, w i+(μ),j = 0, i, j ∈ I, 1 ≤ μ ≤ m}, and the differential ideal is μ
I = I(CR) = {f ∈ F | f = P μ w0(μ) + P w(μ)0 } = {f ∈ F | f = P F }, μ
μ
where differential polynomials P μ , P ∈ F[D, D] and P = (P μ , P ) ∈ F m [D, D]. We consider the form ω = φ dm z ∈ Ωm (F ) with φ = φ(z, w) ∈ F \ I (i.e., ∂z μ φ = 0, ∂wij φ = 0, and φSol(CR) = 0). Here, dω = (Dμ φ)dz μ ∧ dm z, where Dμ φ =
(∂wij φ)wi,j+(μ) = p(D, D)w0,(μ) = p(D, D)Fμ ,
i,j
p(D, D) =
(∂wij φ)Di Dj ∈ F[D, D],
i,j
and hence dω = P μ (D, D)Fμ ∈ Ωm+1 (I) and P μ (D, D) = p(D, D)dz μ ∧ dm z. The equivalence class [ω] = ω + Ωm (I) + dΩm−1 (F ) ∈ CLm (CR) is nontrivial, i.e., [ω] = 0, if the variational derivative δw φ =
(−1)|i+j| Di Dj (∂wij φ) = 0
i,j
(cf. Theorem 4; we note that δw φ = 0). Further, dP = (dP μ ) ∈ Ωm+2 (DI), and the equivalence class [dP ] =
dP + dΩm+1 (DI) ∈ H m+2 (DI) is nontrivial if the class [ω] is nontrivial, where dP μ = Dν a p(D, D) dz ν ∧ dz μ ∧ dm z. In particular, if we take φ = w ∈ F, then ω = w dm z, δw w = 1 = 0, dω = w0,(μ) dz μ ∧ dm z, P μ = dz μ ∧ dm z, and dP μ = Dν dz ν ∧ dz μ ∧ dm z. Therefore, 0 = CLm (CR) H m+2 (DI), m < 2m − 1 for m ≥ 2. ∞ 3.7.2. Yang–Mills system. Here, X = Rm , U = Am , U = Am×I , I = Zm + , F = Cfin (XU; R), and ∞ (XU; A), where A is a Lie algebra with a bracket [ · , · ] and a Killing metric ( · , · ), Φ = Cfin
([A, B], C) + (B, [A, C]) = 0 for all A, B, C ∈ A. In more detail, • U = {A = (Aμ ) | Aμ ∈ A, 1 ≤ μ ≤ m} and • U = {A = (Aμ,i ) | Aμ,i ∈ A, 1 ≤ μ ≤ m, i ∈ I}. Further, 1568
• g = gμν = diag(1, −1, . . . , −1) is the Minkowski metric on Rm ; g −1 = g μν = g; • Dλ ∈ EndR (F ), EndR (Φ), 1 ≤ λ ≤ m, are the total derivatives, Dλ Aμ,i = Aμ,i+(λ) ; • ∇λ ∈ EndR (Φ), 1 ≤ λ ≤ m, are the covariant derivatives; ∇λ f = Dλ f + [Aλ , f ], f ∈ Φ; • F = Fμν dxμ ∧ dxν is the curvature form, where 1 ≤ μ, ν ≤ m; Fμν = [∇μ , ∇ν ] = Dμ Aν − Dν Aμ + [Aμ , Aν ] ∈ EndF (Φ);
• L = (Fμν , F μν )/4 is the Yang–Mills Lagrangian, F μν = g μμ g νν Fμ ν . The Yang–Mills system YM = 0 is the Euler–Lagrange system for the Lagrangian L, i.e., YM = (YM ν ),
YM ν = δAν L = ∇μ F μν ∈ Φ,
1 ≤ ν ≤ m.
Hence, Sol(YM ) = {(x, A) ∈ XU | Di YM ν = 0, i ∈ I, 1 ≤ ν ≤ m}, I = I(YM ) = f ∈ F | f = (P (D), YM ) = (Pν (D), YM ν ) , where Pν (D) =
fνi Di ∈ Φ[D],
(Pν (D), YM ν ) =
i
(fνi , Di YM ν ).
i
There is a differential identity, namely, ∇ν YM ν = 0. Indeed, 1 ∇ν YM ν = ∇ν ∇μ F μν = − [∇μ , ∇ν ]F μν = [Fμν , F μν ] = 0. 2 We consider the 0-form ω = L ∈ Ω0 (F ) = F . Here, dω = Dλ L dxλ , where
1 Dλ (Fμν , F μν ) = (Dλ Fμν , F μν ) + (Fμν , Dλ F μν ) = 4 1
= (Dλ Fμν + [Aλ , Fμν ], F μν ) + (Fμν , Dλ F μν + [Aλ , F μν ]) = 4 1 1
= (∇λ Fμν , F μν ) + (Fμν , ∇λ F μν ) = (Fμν , ∇λ F μν ), 1 ≤ λ ≤ m. 4 2
Dλ L =
Now let m = 2. Then • Fμν = eμν F , where eμν is the sign of the permutation (1, 2) → (μ, ν); F = F12 = D1 A2 − D2 A1 + [A1 , A2 ],F μν = −Fμν = Fνμ ; • L = −(F, F )/2, YM 1 = ∇2 F , and YM 2 = −∇1 F ; • D1 L = −(F, ∇1 F ) = (F, YM 2 ) and D2 L = −(F, ∇2 F ) = −(F, YM 1 ); • dL = (F, YM 2 ) dx1 − (F, YM 1 ) dx2 ∈ Ω1 (I). 1569
In more detail, dL = (p1μ , YM μ ) dx1 + (p2μ , YM μ ) dx2 = (P1 , YM ) dx1 + (P2 , YM ) dx2 , where P1 = p11 p12 , P2 = p21 p22 ∈ Φ2 [D], p11 = 0, p12 = F , p21 = −F , and p22 = 0, YM = YM 1 YM 2 t ∈ Φ2 . In other words, dL = (χ, YM ), where the form χ = P1 dx1 + P2 dx2 ∈ Ω1 (Φ2 [D]). It follows from the equality (dχ, YM ) = ddL = 0 that dχ = −D1 a F D2 a F d2 x ∈ Ω2 (DI). Therefore, the equivalence class L = L + I ∈ CL0 (YM ), L = 0, and the equivalence class dχ = dχ + dΩ1 (DI) ∈ H 2 (DI), dχ = 0, i.e., 0 = CL0 (YM ) H 2 (DI),
0 < 2 − 1 = 1.
We note that here (dχ, YM ) = −(F, ∇ν YM ν ) d2 x in accordance with the general formula ∇ν YM ν = 0 above. We let m ≥ 2 and suppose that cen A = {C ∈ A | [C, A] = 0 for all A ∈ A} = 0. We consider the form ω = (1/2)(C, F μν ) dμν x ∈ Ωm−2 (F ), where C ∈ cen A and the (m−2)-forms ˇμ · · · dx ˇ ν · · · ∧ dxm , dμν x = −dνμ x, 1 ≤ μ, ν ≤ m, are defined by the formula dμν x = (−1)μ+ν dx1 ∧ · · · dx μ < ν. Here, dω =
1 1 Dλ (C, F μν ) dμν x = (C, Dλ F μν ) dxλ ∧ dμν x = (C, Dμ F μν ) dν x = 2 2
= (C, ∇μ F μν ) dν x = (C, YM ν ) dν x = (P ν (D), YM )dν x ∈ Ωm−1 (I), where P ν (D) = δλν C | 1 ≤ λ ≤ m ∈ Φm [D]. Hence, the equivalence class ω = ω + dΩm−3 (F )+ Ωm−2 (I) ∈ CLm−2 (YM ), and the equivalence class dχ = dχ + dΩm−1 (DI) ∈ H m (DI), where χ = P ν (D) dν x, dχ = Dν a P ν (D) dm x = CDλ | 1 ≤ λ ≤ m dm x. Unfortunately, we cannot ensure that the equivalence class ω (and hence dχ) is nontrivial. We note that (CDλ , YM λ ) = (C, Dλ YM λ ) = (C, ∇λ YM λ ) = 0.
4. Evolution with constraints 4.1. Evolution systems. We keep the above notation (see Sec. 2.1) and introduce some new notation: • T = R is the space of an additional independent variable t (time); α α • V = RA Z+ ×I = {v = (vp,i ) | vp,i ∈ R, α ∈ A, p ∈ Z+ , i ∈ I} is the extended space of differential α α α variables, and the projection Π : V → U is defined by the rule v = (vp,i ) → u = (uα i ), ui = v0,i ; ∞ ∞ (TXU; R), and F = Cfin (TXV; R); and • Ψ = C ∞ (TX; R), E = Cfin α α α , Dμ = ∂xμ + α • Dt = ∂t + α,p,i vp+1,i ∂vp,i α,p,i vp,i+(μ) ∂vp,i , 1 ≤ μ ≤ m, [Dμ , Dν ] = 0, and [Dt , Dμ ] = 0 for all 1 ≤ μ, ν ≤ m.
An evolution system is written as E = 0, where E = (E α ), α α α E α = uα t − f = v1,0 − f (t, x, u),
1570
α ∈ A,
f = (f α ) ∈ E A .
The solution manifold of this system is α α = fp,i , α ∈ A, p ∈ Z+ , i ∈ I} ∼ Sol(E) = {(t, x, v) ∈ TXV | vp+1,i = TXU, α where fp,i = Dpt Di f α . We note that E ⊂ F. The system E = 0 is therefore regular, and we treat the space TXU as a global chart on the manifold Sol(E). A function ψ ∈ Ψ is a solution of the system E = 0 if ∂t ψ α (t, x) = f α (t, x, u)u=jψ(t,x) , jψ = (ψiα = ∂xi ψ α ) ∈ ΨA I .
Because the system E = 0 is regular, the differential ideal I(E) = f ∈ F | f Sol(E) = 0 . In addition, we have the quotient algebra F ∼ = E, and the differential algebra (F, D) ∼ = (E, D), where D = (Dt , D1 , . . . , Dm ), D = (Dt , D1 , . . . , Dm ), α uα 1 ≤ μ ≤ m, Dμ = Dμ E = ∂xμ + i+(μ) ∂ui , α,i
Dt = Dt E = ∂t + fiα ∂uαi ,
fiα = Di f α .
α,i
Hence, {Ωq (F); d } ∼ = {Ωq (E); dq }, and H q (F) = H q (E), q ∈ Z, where q
d = dt ∧ Dt + dxμ ∧ Dμ ,
d = dt ∧ Dt + dxμ ∧ Dμ .
We consider the complex {Ωqx (E); dqx }, where 1 q Ωx = ω = ωμ1 ,...,μq dxμ1 ∧ · · · ∧ dxμq q!
ωμ1 ...μq ∈ E ,
dx = dxμ ∧ Dμ
(the variable t ∈ T here is a parameter). We then obtain the representation ω = ω + dt ∧ ω ,
Ωq (E) = Ωqx (E) ⊕ dt ∧ Ωq−1 x (E),
q where ω = πt ω = ιDt ω ∈ Ωq−1 x (E) and ω = πx ω = ω − dt ∧ ω ∈ Ωx (E), q ∈ Z, and the differential d = dt + dx ,
dt = dt ∧ Dt : Ωqx (E) → dt ∧ Ωqx (E),
dx = dxμ ∧ Dμ : Ωqx (E) → Ωq+1 x (E).
This representation generates the exact sequence of complexes π
ε
x q−1 0 → {Ωq−1 → {Ωq (E); dq } −→ {Ωqx (E); dqx } → 0, x (E); dx } −
which is written as the commutative diagram of linear spaces with exact rows presented in Fig. 2, where ε = εq : Ωqx (E) → Ωq+1 (E) and ω → ω = (−1)q dt ∧ ω . In particular, the exact sequence of linear spaces of cohomologies 0
π
Δ
π
Δ
ε
x H 2 (E) −− → Hx2 (E) −→· · ·
Δ
Hx1 (E) −→ ε
· · · −→Hxm−1 (E) −→ Δ
Δ
x H 1 (E) −− → Hx1 (E) −→
Hx0 (E) −→
−→
π
ε
Δ
Δ
Δ
x H 0 (E) −− → Hx0 (E) −→
−→ −→
π
−→
ε
x m H m (E) −− →H → x (E) −
π
x Hxm (E) −→H m+1 (E) −− →
0,
is defined, where Δ = Δq : Hxq (E) → Hxq (E) and ω → Δω = Dt ω . 1571
0
0
0
ε-
? Ω0 (E)
? πx Ω0x (E)
- 0
0
dx ? - Ω0 (E) x
ε-
d ? Ω1 (E)
dx ? πx Ω1x (E)
- 0
0
dx ? - Ω1x (E)
ε-
d ? Ω2 (E)
dx ? πx Ω2x (E)
- 0
d ? .. .
dx ? .. .
0
? - 0
dx ? .. .
0
0
dx d ? ? ε - Ωm−1 (E) - Ωm (E) x dx ? - Ωm x (E) ? 0
dx ? πx Ωm x (E)
- 0
dx d ? ? ε π x - Ωm+1 (E) - 0
- 0
? 0
? 0
Fig. 2
Theorem 8. For any evolution system E = 0, the spaces of conservation laws CLq (E) = 0 for all 0 ≤ q ≤ m − 1. Proof. It suffices to use the equalities H q (F) = H q (E), the above exact sequence, and Theorem 4. Therefore, an evolution system E = 0 can have only standard conservation laws CLm (E) = H m (E) = Ker dm / Im dm−1 , where Ker dm = {ω = ρdm x + dt ∧ J μ dμ x | ρ, J μ ∈ E, Dt ρ − Dμ J μ = 0}, Im dm−1 = {ω = dη | η = j μ dμ x + dt ∧ aμν dμν x, j μ , aμν ∈ E}, and hence
Ker dm {(ρ, J) ∈ E × E m | Dt ρ − Dμ J μ = 0} . = Im dm−1 {(ρ, J) ∈ E × E m | ρ = Dμ j μ , J μ = Dt j μ + Dν aμν , j μ , aμν ∈ E}
Theorem 9. For an evolution system E = 0, the linear space of conservation laws CLm (E) =
{ρ ∈ E | δuα (Dt ρ) = 0, α ∈ A} = {ρ ∈ E | δuα ρ = 0, α ∈ A}
= {χ ∈ EA | χ∗ = χ∗ , Dt χ + f ∗ χ = 0}, 1572
where χ = (χα ), χα = δuα ρ, α ∈ A, f∗ :
EA → EA ,
(f ∗ χ)α =
(−1)|i| Di (χβ ∂uαi f β ),
β,i
χ∗ :
E A → EA , (χ∗ g)α =
g = (g β ) ∈ E A ,
(∂uβ χα )Di g β , i
β,i
χ∗ :
E A → EA , (χ∗ g)α =
(−1)|i| Di (g β ∂uαi χβ ),
g = (g β ) ∈ E A .
β,i
Proof. The proof can be found, for example, in [1].
4.2. Evolution with constraints. We keep all above notation. An evolution system with constraints EF = 0 consists of two subsystems: an evolution system E = 0 and a system of constraints F = 0, where α E = (E α = uα t − f ),
f = (f α ) ∈ E A ,
F = (F κ ) ∈ E K .
We suppose that these subsystems are compatible. Namely, let J = J (F ) = {φ = P (D)F ∈ E | P (D) ∈ EK [D]} be the differential ideal of the differential algebra (E, D), D = (D1 , . . . , Dm ), generated by the constraints system F = 0, (we note that (F , D) ⊂ (E, D)). Then the compatibility condition becomes Dt J : J → J , i.e., Dt F = Q(D)F ∈ J K . Now, Sol(EF ) = Sol(E) ∩ Sol(F ) ∼ = ∼ = {(t, x, u) ∈ TXU | Di F κ (t, x, u) = 0, i ∈ I, κ ∈ K}, and the spaces of conservation laws (see Sec. 3.4) CL0 (EF ) = H 0 (E)/R,
CLq (EF ) = H q (E),
1 ≤ q ≤ m,
where E = E J . As above, the exact sequence of linear spaces 0 → J → E → E → 0 generates a chain sequence of differential complexes (cf. Fig. 1) q
0 → {Ωq (J ); dq } → {Ωq (E); dq } → {Ωq (E); d } → 0 and consequently an exact sequence of linear spaces of cohomologies Δ
0 −→ H 0 (J ) −→ H 0 (E) −→ H 0 (E) −→ Δ
Δ
Δ
Δ
−→ H 1 (J ) −→ H 1 (E) −→ H 1 (E) −→ · · · · · · −→ H m (J ) −→ H m (E) −→ H m (E) −→ Δ
−→ H m+1 (J ) −→ H m+1 (E) −→ H m+1 (E) −→ 0, where the connecting linear maps Δ = Δq : H q (E) → H q+1 (J ),
ω → Δω,
0 ≤ q ≤ m − 1,
are defined by the rule Δω = dω for all ω ∈ Ωq (E) and dω ∈ Ωq+1 (J ). 1573
Theorem 10. For any system EF = 0, we have the isomorphisms • Δ : CLq (EF ) H q+1 (J ),
0 ≤ q ≤ m − 2,
• Δ : CLm−1 (EF ) H m (J ) ∩ dΩm−1 (E). Proof. It suffices to use the above exact sequence of linear spaces and Theorem 8. We note that H 0 (J ) = 0. Theorem 11. For every system EF = 0, we have the representation CLm (EF )
{χ ∈ EA | χ∗ = χ∗ , Dt χ + f ∗ χ = δu (φF ), φ ∈ EK } . {χ = δu (ψF ) | ψ ∈ EK }
Proof. The proof is an appropriate modification of the proof of Theorem 9. We note that δu (φF ) = φ F + F ∗ φ and δu (ψF ) = ψ ∗ F + F ∗ ψ. ∗
There is another way to treat the system EF = 0. Namely, the representation Ωq (E) = Ωqx (E) ⊕ dt ∧ Ωq−1 x (E),
ω = ω + dt ∧ ω ,
d = dt + dx ,
generates the exact sequence of complexes (cf. Fig.2) q−1
0 → {Ωq−1 x (E); dx
q
ε
q
π
x }− → {Ωq (E); d } −−→ {Ωqx (E); dx } → 0
and yields the following theorem. Theorem 12. The exact sequence of linear spaces 0
π
Δ
x −→ H 0 (E) −−→ Hx0 (E) −→
Δ
ε
π
Δ
Δ
ε
π
Δ
Δ
ε
π
Δ
Δ
ε
π
x −→ Hx0 (E) −→ H 1 (E) −−→ Hx1 (E) −→ x −→ Hx1 (E) −→ H 2 (E) −−→ Hx2 (E) −→ · · · x Hxm (E) −→ · · · −→ Hxm−1 (E) −→ H m (E) −−→ x −→ Hxm (E) −→ H m+1 (E) −−→
0
is defined, where Δ = Δq : Hxq (E) → Hxq (E) and ω → Δω = ω = Dt ω . Proof. Theorems 5, 6 and 7 can be used to calculate the spaces Hxq (E). 4.3. Examples. 4.3.1. Parabolic constraint. We consider the system EF = 0, where E = ut − uxy ,
F = uy − uxx ,
and the form ω = u dx + ux dy + uy dt ∈ Ω1 (E). The differential dω = (uy dy + ut dt) ∧ dx + (uxx dx + uxt dt) ∧ dy + (uxy dx + uuu dy) ∧ dt = = (−uy + uxx ) dx ∧ dy + (ut − uxy ) dt ∧ dx + (uxt − uyy ) dt ∧ dy = = −F dx ∧ dy + dt ∧ (E dx + (Dx E − Dy F ) dy) ∈ Ω2 (J ). 1574
Hence, the equivalence class ω ∈ CL1 (EF ), and it is easy to verify that ω = 0. Further,
0 = ddω = −Dt F + Dy E − Dx (Dx E − Dy )F dt ∧ dx ∧ dy =
= (Dy − Dx Dx )E − (Dt − Dx Dx )F dt ∧ dx ∧ dy, i.e., the equivalence class P (D) = PF , PE ∈ H 3 (DI), where PF = (Dy − Dx Dx ) ⊗ dt ∧ dx ∧ dy,
PE = −(Dt − Dx Dy ) ⊗ dt ∧ dx ∧ dy.
4.3.2. Elliptic constraint. We consider the system EF = 0, E = ut − f(ξ, η),
F = uxx + uyy ,
ξ = ux ,
η = uy ,
where f(ξ, η) is a harmonic function (i.e., fξξ + fηη = 0). Let A, B, C(ξ, η) be smooth functions satisfying the equations ⎧ ⎧ ⎨Cξ = Aξ fξ − Aη fη , ⎨Aξ = Bη , ⎩A = −B , η ξ
⎩C = B f − B f . η η η ξ ξ
We note that by virtue of the first subsystem, the function W (ζ) = A(ξ, η) + iB(ξ, η) is homomorphic, ζ = ξ + iη, while the assumption fξξ + fηη = 0 is the compatibility condition for the second subsystem. We now consider the form ω = A dx + Bdy + C dt ∈ Ω1 (E). Here, the differential
dω = (Dx B − Dy A) dx ∧ dy + dt ∧ (Dt A − Dx C) dx + (Dt B − Dy C) dy =
= Bξ F dx ∧ dy + dt ∧ (Aξ Dx E + Aη Dy E + Aη fη F ) dx + + (Bξ Dx E + Bη Dy E + Bξ fξ F ) dy ∈ Ω2 (J ), and hence the equivalence class ω ∈ CL1 (EF ). It can be verified that ω = 0 iff A = λux , B = λuy , and C = λf for some λ ∈ R. Further,
0 = ddω = Dt (Bξ F ) − Dx (Bξ Dx E + Bη Dy E + Bξ fξ F ) + + Dy (Aξ Dx E + Aη Dy E + Aη fη F ) dt ∧ dx ∧ dy = = PE (D)E + PF (D)F ∈ Ω3 (E), where
PE (D) = −Dx a (Bξ Dx + Bη Dy ) + Dy a (Aξ Dx + Aη Dy ) ⊗ dt ∧ dx ∧ dy,
PF (D) = Dt a Bξ − Dx a Bξ fξ + Dy a Aη fη ⊗ dt ∧ dx ∧ dy, and hence P (D) = PE (D), PF (D) ∈ Ω3 (DI), and P (D) ∈ H 3 (DI). 1575
4.3.3. Hyperbolic constraint. We consider the system EF = 0, E = ut − f(ξ, η),
F = uxx − uyy ,
ξ = ux ,
η = uy ,
where fξξ − fηη = 0. Let A, B, C(ξ, η) be smooth functions satisfying the equations ⎧ ⎨Aξ = Bη ,
⎧ ⎨Cξ = Aξ fξ + Aη fη ,
⎩A = B , η ξ
⎩C = B f + B f . η ξ ξ η η
We consider the form ω = A dx + Bdy + C dt ∈ Ω1 (E). The differential
dω = (Dx B − Dy A) dx ∧ dy + dt ∧ (Dt A − Dx C) dx + (Dt B − Dy C) dy =
= Bξ F dx ∧ dy + dt ∧ (Aξ Dx E + Aη Dy E − Aη fη F ) dx + + (Bξ Dx E + Bη Dy E + Bξ fξ F ) dy ∈ Ω2 (J ), and hence the equivalence class ω ∈ CL1 (EF ). It can be verified that ω = 0 iff A = λux , B = λuy , and C = λf for some λ ∈ R. Further,
0 = ddω = Dt (Bξ F ) − Dx (Bξ Dx E + Bη Dy E + Bξ fξ F ) + + Dy (Aξ Dx E + Aη Dy E − Aη fη F ) dt ∧ dx ∧ dy = = PE (D)E + PF (D)F ∈ Ω3 (E), where
PE (D) = −Dx a (Bξ Dx + Bη Dy ) + Dy a (Aξ Dx + Aη Dy ) ⊗ dt ∧ dx ∧ dy, PF (D) = (Dt a Bξ − Dx a Bξ fξ − Dy a Aη fη ) ⊗ dt ∧ dx ∧ dy, and hence P (D) = PE (D), PF (D) ∈ Ω3 (DI) and P (D) ∈ H 3 (DI). 4.4. Continuity equations. In some applications, the constraint system F = 0 is the system of continuity equations, F = (F κ ) ∈ E K , F κ = ∂xμ uκμ . In this case, the cohomology spaces Hxq (E) can be calculated explicitly. For this, we rewrite the system ∂xμ uκμ = 0 as ∂x1 uκ1 + 2≤α≤m ∂xα uκα = 0 and treat it as an evolution system, G = ∂s v − g = 0,
v = (v κ , v κα ) ∈ V,
g = (g k , gικα ) ∈ F A ,
where we use the local notation • s ∈ S = R, y = (y 2 , . . . , y m ) ∈ Y = Rm−1 , s = x1 , y α = xα , 2 ≤ α ≤ m; • A = K ∪ B, B = {(κ, α, ι) | κ ∈ K, 2 ≤ α ≤ m, ι ∈ Z+ }; , i = (ι, j) ∈ I; • J = {j = (j α ) | j α ∈ Z+ , 2 ≤ α ≤ m} = Zm−1 + 1576
• V = {v = (v κ , vικα ) | κ ∈ K, (κ, α, ι) ∈ B} = RA , κα κ = vι0 = uκα v κ = v0κ = uκ1 = uκ1 0 , vι (ι,0) ; κα ) | κ ∈ K, (κ, α, ι) ∈ B, j ∈ J}, • V = {v = (vjκ , vιj κα κα vjκ = uκ1 (0,j) , vιj = u(ι,j) ; ∞ • F = Cfin (SY V; R);
κβ ∂vκβ , • Ds = ∂s + j∈J gjκ ∂vjκ + gιj ιj κ κβ Dα = ∂yα + j∈J vj+(α) ∂vjκ + vι,j+(α) ∂vκβ , 2 ≤ α ≤ m; ιj
κβ κβ κβ κβ κβ • g κ = −v0,(β) , gικβ = vι+1,0 , gjκ = Dj g κ = −v0,j+(β) , gιj = Dj gικβ = vι+1,j .
By Theorem 8, we have H 0 (F ) = R,
H q (F ) = 0,
1 ≤ q ≤ m − 2,
and by Theorem 9, H m−1 (F ) {χ ∈ FA | χ∗ = χ∗ , Ds χ + g∗ χ = 0},
[ω] → χ = δv J s ,
where ω = J s dm−1 y + J α ds ∧ dα y and Ds J s + Dα J α = 0. Here, g∗ : F A → F A ,
κ κα φ = (φκ , φκα ι ) → g∗ φ = ((g∗ φ) , (g∗ φ)ι ),
κα κα where (g∗ φ)κ = −Dβ φκβ 0 , (g∗ φ)ι = φι+1 , and
g ∗ : FA → FA ,
χ = (χκ , χικα ) → g∗ χ = ((g∗ χ)κ , (g∗ χ)ικα ),
ι −1 ∗ where (g∗ χ)κ = 0, (g∗ χ)ικα = χι−1 κα + δ0 Dα χκ (we note that χκα = 0). Further, the equation Ds χ + g χ = 0 reduces to the system
Ds χ = 0,
Ds χ0κα + Dα χκ = 0,
Ds χικα + χι−1 κα = 0
and has the general solution χ = (χκ , 0), χκ ∈ R, κ ∈ K (the crucial fact here is that every function f ∈ F has a finite order because it by definition depends on a finite number of variables; see, e.g., [3], [8], [12], [13] for more details). The condition χ∗ = χ∗ is trivially satisfied. We return to the our global notation. Theorem 13. The system F = 0, where F = (F κ ) ∈ F K , F κ = ∂xμ uκμ , κ ∈ K, and F = C ∞ (XU; R), has the spaces of conservation laws
CLq (F ) =
⎧ ⎨0,
0 ≤ q ≤ m − 2,
⎩{ω = χ eκ | χ ∈ R} R , q = m − 1, κ κ K
where eκ = uκμ dμ x.
Proof. The proof follows from the above discussion and the trivial equality δvκ v κ =δκκ , κ, κ ∈ K. 1577
Remark 1. It is clear that every current ω = χκ uκμ , χκ ∈ R, here defines a nontrivial conservation law. The above theorem states that all conservation laws have this form. Corollary 2. The constraint system F = 0, where F = (F κ ) ∈ E K , F κ = ∂xμ uκμ , κ ∈ K, and E = C ∞ (TXU; R), has the cohomology spaces ⎧ ⎪ ⎪ ⎪T , ⎨
Hxq (E)
=
⎪ ⎪ ⎪ ⎩
q = 0, 1 ≤ q ≤ m − 2,
0,
{ω = χκ eκ | χκ ∈ T } TK , q = m − 1,
where T = C ∞ (T; R) and eκ = uκμ dμ x.1 4.5. Zero divergence constraint. An evolution with the zero-divergence constraint is described by the system EF = 0, where κμ E = (E κμ = uκμ ), t −f
f = (f κμ ) ∈ E K×m ,
F = (F κ = ∂xμ uκμ ) ∈ E K ,
i.e., here A = K × m and U = RK×m . We suppose that the subsystems E = 0 and F = 0 are compatible, I i.e., Dt F κ = Dt Dμ uκμ = Dμ Dt uκμ = Dμ f κμ ∈ J , κ ∈ K. By Theorem 12 and Corollary 2, the exact sequences H 0 (E) = Ker ΔT ,
Δ
0 → H 0 (E) → T −→ T , T −→ T → H 1 (E) → 0,
H 1 (E) = T Im ΔT ,
0 → H q (E) → 0,
H q (E) = 0,
Δ
π
Δ
x 0 → H m−1 (E) −−→ Hxm−1 (E) −→ Hxm−1 (E),
2 ≤ q ≤ m − 2,
H m−1 (E) = Ker Δm−1 ,
are defined, where Δm−1 = ΔH m−1 (E) : Hxm−1 (E) → Hxm−1 (E) and Δm−1 [ω] = [Dt ω]. We can now calculate x
Ker ΔT = R,
Im ΔT = T ,
Ker Δm−1 = {ω = χκ eκ | χκ ∈ T , ∂t χκ + χσ Lσκ = 0} RK , where by the compatibility condition, [Dt eσ ] = [(Dt uσμ )dμ x] = [f σμ dμ x] = Lσκ eκ ,
Lσκ = δuκ1 f σ1 ∈ T .
We hence have the following theorem. Theorem 14. The spaces of conservation laws of the above system EF = 0 are
CLq (EF ) = 1 We
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⎧ ⎨0,
0 ≤ q ≤ m − 2,
⎩{ω = χ eκ | χ ∈ T , ∂ χ + χ Lσ = 0} R , q = m − 1. κ κ t κ σ κ K
note that the variable t ∈ T here is a parameter.
Remark 2. The space CLm−1 (EF ) has yet another (more natural) representation. Namely, according to the equality H m−1 (E) = Ker Δm−1 , for every basic form eκ ∈ Ker Δm−1 , κ ∈ K, there exists the basic class [J κ ] ∈ H m−1 (E) such that π x [J κ ] = eκ , where J κ = eκ + dt ∧ ψ κ ∈ Ωm−1 (E), dJ κ ∈ Ωm (J ), (E), and dx ψ κ − Dt eκ ∈ Ωm−1 (J ). Hence, ψ κ ∈ Ωm−2 x CLm−1 (EF ) = {χκ [J κ ] | ∂t χκ + χσ Lσκ = 0}. Remark 3. We note that evolution systems with a controlled divergence κμ E κμ = uκμ = 0, t −f
F κ = ∂xμ uκμ − ρκ = 0,
where ρκ ∈ C ∞ (TX; R) ⊂ E, reduce to systems with a zero divergence ˜ κμ = vtκμ − gκμ = 0, E
F˜ κ = ∂xμ v κμ = 0
by the change of variables v κμ = uκμ − ϕκμ ,
gκμ (t, x, v) = f κμ (t, x, u) − ∂t ϕκμ ,
where ϕ(t, x) is a particular solution of the system F = 0, i.e., ∂xμ ϕκμ − ρκ = 0. 4.6. Example: Maxwell’s equations. In the standard physical notation where • t ∈ T = R and x ∈ X = R3 are time and space variables, • E, H ∈ V = R3 are the electric and the magnetic fields, and • ρ(t, x) and j(t, x) are the charge and the current density, Maxwell’s equations have the form ∂t H + rot E = 0,
div H = 0,
∂t E − rot H + j = 0,
div E − ρ = 0.
The compatibility condition here is written as ∂t ρ + div j = 0 and yields the representation ρ = div R, j = −∂t R + rot S with some vector functions R(t, x) and S(t, x). This allows rewriting Maxwell’s equations in the zero-divergence form: ∂t H + rot E = 0,
div H = 0,
∂t (E − R) − rot(H − S) = 0,
div(E − R) = 0.
Hence, here K = {1, 2}, m = 3, u1λ = H λ ,
u2λ = E λ − Rλ ,
1 ≤ λ ≤ 3,
f 1λ = − rot Hλ = −eλμν ∂xμ Hν ,
f 2λ = rot(E − R)λ = eλμν ∂xμ (Eν − Rν ),
F 1 = div H = ∂xμ H μ ,
F 2 = div(E − R) = ∂xμ (E μ − Rμ ) 1579
(in the Euclidean space, the components Vμ = δμν V ν = V μ for any vector V). Further, e1 = H μ dμ x,
dx e1 = dx H μ dμ x = (div H) dm x ∈ Ωm x (J ),
e2 = (E μ − Rμ ) dμ x,
dx e2 = dx (E μ − Rμ ) dμ x = div(E − R) dm x ∈ Ωm x (J ).
Now, δuκ1 f σ1 = 0, κ, σ ∈ K. Hence, Lσκ = 0, and the system ∂t χκ + χσ Lσκ = 0 has the general solution χκ ∈ R. Finally, we can take J 1 = H μ dμ x − dt ∧ Eμ dxμ ,
J 2 = (E μ − Rμ ) dμ x + dt ∧ (Hμ − Sμ ) dxμ
because in this case, π x J κ = eκ , κ ∈ K, and dJ 1 = (div H) dm x + dt ∧ (∂t H + rot E)μ dμ x = 0,
μ dJ 2 = div(E − R) dm x + dt ∧ ∂t (E − R) − rot(H − S) dμ x = 0 on solutions of Maxwell’s equations. Therefore, according to the above discussion, we have the representations CLm−1 (EF ) = {χκ eκ | χκ ∈ R} = {χκ J κ | χκ ∈ R} R2 . Maxwell’s equations have a nice description in the language of differential forms. Namely, the electromagnetic field tensor F is here written as a 2-form F = H μ dμ x − Eμ dt ∧ dxμ = J 1 , its Hodge dual ∗F is the 2-form ∗F = E μ dμ x + dt ∧ Hμ dxμ = J 2 + (Rμ dμ x + Sμ dt ∧ dxμ ), and Maxwell’s equations are written as (see, e.g., [14]) dF = dJ 1 = 0,
d(∗F ) − (ρ d3 x − j μ dt ∧ dμ x) = dJ 2 = 0.
We note that d(Rμ dμ x + Sμ dt ∧ dμ ) = ρ d3 x − j μ dt ∧ dμ x). Hence, Maxwell’s equations are written as two-dimensional conservation laws in this setting. According to Theorem 14, Maxwell’s equations do not admit any other low-dimensional conservation laws.
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