VARIATIONAL FIRST-ORDER QUASILINEAR EQUATIONS

on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary

VARIATIONAL FIRST-ORDER QUASILINEAR EQUATIONS

JOSEPH GRIFONE, J. MU ˜NOZ MASQU ´E, AND L.M. POZO CORONADO

Abstract. The systems of first-order quasilinear partial differential equations defined on the 1-jet bundle of a fibred manifold which come from a variational problem defined by an affine Lagrangian are characterized by means of the Hamilton–Cartan formalism and the theory of formal integrability.

1. Introduction

The goal of this paper is to analyse the role that Hamiltonian formalism and formal integrability play in studying the variational character of first-order quasi- linear partial differential systems.

Letp:M →N be a fibred manifold over an orientable connectedC^∞ manifold N. Set dimN =n, dimM =m+n. Letp1:J¹→N be the 1-jet bundle of local sections of p, and let p10: J¹M → J⁰M = M denote the canonical projection:

p10(j_x¹s) =s(x). Throughout this paper Greek indices run from 1 tom, and Latin indices run from 1 ton.

If (xⁱ, y^α) is a fibred coordinate system for the submersionpdefined on an open subsetU ⊆M, we denote by (xⁱ, y^α;y_i^α) the coordinate system induced onJ¹U in a natural way;i.e.,y_i^α(j_x¹s) =∂(y^α◦s)/∂xⁱ(x).

The starting point is the following basic fact:

Proposition 1. LetLvbe a Lagrangian density onJ¹M, wherevis a volume form onN. The Poincar´e–Cartan formΘofLv isp10-projectable if and only ifLis an affine function, or in other words, locally there exist functions A⁰, Aⁱ_α ∈ C^∞(M) such that L=A⁰+Aⁱ_αy_i^α. In this case, the Euler–Lagrange equations of Lv are a system of first-order quasilinear equations on J¹M.

Proof. It is an easy consequence of the local expression of the Poincar´e–Cartan form

(1) Θ = (−1)ⁱ⁻¹∂L

∂y_i^αθ^α∧vi+Lv,

1991Mathematics Subject Classification. Primary 37K05; Secondary 58A20, 58E30.

Key words and phrases. Affine Lagrangian, involutiveness, Poincar´e–Cartan form.

Supported by CICYT (Spain) grant PB98–0533.

131

associated with Lv, where we have chosen the fibred coordinate system (xⁱ, y^α) such that,

v=dx¹∧. . .∧dxⁿ, v_i=dx¹∧. . .∧dxcⁱ∧. . .∧dxⁿ,

and θ^α are the standard 1-contact forms onJ¹M; i.e., θ^α=dy^α−y_i^αdxⁱ. Then, as a simple calculation shows, the Euler–Lagrange equations are:

∂A⁰

∂y^α −∂Aⁱ_α

∂xⁱ =∂Aⁱ_α

∂y^β −∂Aⁱ_β

∂y^α ∂y^β

∂xⁱ, where the indexαis free.

This result poses the problem of determining which systems of first-order quasi- linear equations onJ¹M come from an affine Lagrangian as above. The character- ization is as follows:

Theorem 2. With the previous hypotheses and assumptions, the system of C^ω equations

(2) Fα=F_αβⁱ ∂y^β

∂xⁱ, F_αβⁱ =F_βαⁱ , Fα, F_αβⁱ ∈C^∞(M),

is variational with respect to an affine Lagrangian if and only if the differential formξ∈Ωⁿ⁺¹(M)defined by

(3) ξ=Fαdy^α∧v+ (−1)ⁱF_αβⁱ dy^α∧dy^β∧vi

is closed; that is, if and only if

(4)

F_αβⁱ +F_βαⁱ = 0,

∂F_αβⁱ

∂xⁱ −∂F_β

∂y^α +∂F_α

∂y^β = 0,

∂F_αβⁱ

∂y^γ +∂F_βγⁱ

∂y^α +∂F_γαⁱ

∂y^β = 0.















In this case, ξ is the exterior differential of the Poincar´e–Cartan form associated with the corresponding Lagrangian densityLv.

The result can also be formulated by saying that a (n−1)-horizontal (overN) differential (n+ 1)-form onM is variational if and only if it is closed.

In the particular casen= dimN= 1, that is, for ordinary differential equations, the result of Theorem 2 was stated in [3]. Moreover, the conditions (4) agree with those obtained in [2, I-VII] in the case of an affine Lagrangian although in the present work the result is obtained by a completely different method.

2. Proof of Theorem 2

Let us first consider the Lagrangian L = A⁰+Aⁱ_αy^α_i. The associated Euler–

Lagrange equations are

∂A⁰

∂y^α −∂Aⁱ_α

∂xⁱ =∂Aⁱ_α

∂y^β −∂Aⁱ_β

∂y^α ∂y^β

∂xⁱ,

and hence the associated differential formξreads ξ=∂A⁰

∂y^α −∂Aⁱ_α

∂xⁱ

dy^α∧v+ (−1)ⁱ∂Aⁱ_α

∂y^β −∂Aⁱ_β

∂y^α

dy^α∧dy^β∧vi,

which coincides with the exterior differential of the Poincar´e–Cartan form and ξ is obviously closed. In fact, in this case from the formula (1) we obtain Θ = A⁰v+ (−1)ⁱ⁻¹Aⁱ_αdy^α∧vi, and henceξ=dΘ.

Conversely, let us suppose that the equations (4) hold, so that the differential form ξin (3) is closed. Then, the crucial point of the result is to prove that there exists a local primitive form ζ ∈ Ωⁿ(M), ξ =dζ, which, in addition, is (n−1)- horizontal with respect top;i.e.,iY₀iY₁ω= 0 for allp-vertical tangent vectorsY0, Y1onM. This means thatζ is written as

(5) ζ=A⁰v+ (−1)ⁱ⁻¹Aⁱ_αdy^α∧vi, A⁰, Aⁱ_α∈C^ω(M).

Therefore, the equation ξ = dζ has a local solution if and only if the following system of PDEs is integrable:

F_α=∂A⁰

∂y^α −∂Aⁱ_α

∂xⁱ , (6)

F_αβⁱ =∂Aⁱ_α

∂y^β −∂Aⁱ_β

∂y^α. (7)

Theorem 3. If the conditions (4) hold, then the system (6,7) is involutive and, hence, formally integrable.

Proof. Let us denote by ∧ⁿ₂T^∗M (resp. ∧ⁿ⁺¹₃ T^∗M) the subbundle of ∧ⁿT^∗M (resp. ∧ⁿ⁺¹₃ T^∗M) defined by iY₀iY₁ω = 0 (resp. iY₀iY₁iY₂ω = 0) for allp-vertical tangent vectorsY0, Y1 (resp. Y0, Y1, Y2) in M. Let

Φ :J¹(∧ⁿ₂T^∗M)→ ∧ⁿ⁺¹₃ T^∗M be the affine bundle morphism given by

Φ(j_y¹ζ) = (dζ)y−ξy. SetR₁= ker(Φ,0).

We introduce coordinates (xⁱ, y^α, z⁰, z_αⁱ) (resp. (xⁱ, y^α, w_α, wⁱ_αβ)) in ∧ⁿ₂T^∗M (resp. ∧ⁿ⁺¹₃ T^∗M) as follows

ζ=z⁰(ζ)v+z_αⁱ(ζ)dy^α∧vi,

η =w_α(η)dy^α∧v+wⁱ_αβ(η)dy^α∧dy^β∧v_i.

Let (xⁱ, y^α, z⁰, z_αⁱ;z_i⁰, z_α⁰, z_α,jⁱ , z_α,βⁱ ) denote the system of coordinates induced in J¹(∧ⁿ₂T^∗M); precisely,

z⁰_i(j_y¹ζ) = ∂(z⁰◦ζ)

∂xⁱ (y), z_α⁰(j_y¹ζ) =∂(z⁰◦ζ)

∂y^α (y), z_α,jⁱ (j_y¹ζ) = ∂(z_αⁱ ◦ζ)

∂x^j (y), z_α,βⁱ (j_y¹ζ) =∂(z_αⁱ ◦ζ)

∂y^β (y).

Then, the equations of Φ are

(8) w_α◦Φ =z_α⁰−z_α,iⁱ −F_α, w_αβⁱ ◦Φ =zⁱ_α,β−z_β,αⁱ −F_αβ. Hence Φ has constant rank andR₁ is a fibred submanifold ofJ¹(∧ⁿ₂T^∗M).

Moreover, a section ζof the vector bundle∧ⁿ₂T^∗M satisfies the equations (6,7) if and only ifj¹_yζ∈R1 at every pointy∈M; that is,ζis a solution ofR1. Lemma 4. With the above notations, the vectors

u¹=∂/∂x¹, . . . , uⁿ=∂/∂xⁿ, v¹=∂/∂y¹, . . . , v^m=∂/∂y^m constitute a quasiregular basis for R1 at each point ofM.

Proof (of Lemma). The symbol of Φ,

σ1=σ1(Φ) : T^∗M ⊗ ∧ⁿ₂T^∗M → ∧ⁿ⁺¹₃ T^∗M

is given by σ₁(ω ⊗ζ) = ω ∧ζ, for every ω ∈ T^∗M, ζ ∈ ∧ⁿ₂T^∗M, or in local coordinates

σ1(dx^j⊗v) = 0, σ1(dy^β⊗v) =dy^β∧v,

σ₁(dx^j⊗(dy^α∧v_i)) = (−1)^jδ^j_idy^α∧v, σ1(dy^β⊗(dy^α∧vi)) =dy^β∧dy^α∧vi.

Setg₁= kerσ₁. In order to calculate dimg₁ we first notice that an element χ=λjdx^j⊗v+λⁱ_jαdx^j⊗dy^α∧vi+µβdy^β⊗v+µⁱ_βαdy^β⊗dy^α∧vi

belongs tog1 if and only if (−1)ⁱλⁱ_iα+µα= 0, µⁱ_αβ+µⁱ_βα= 0. Hence dimg1=n+n²m+¹₂nm(m+ 1) =n(1 +nm) +¹₂nm(m+ 1).

Now we must count the dimension of the spaces

g_1,u1,...,u^k ={χ∈g1: iu¹χ=· · ·=i_ukχ= 0}. We observe that g_1,u1 =

χ∈g₁: λ₁=λⁱ_1α= 0 , and hence

dimg_1,u1= dimg₁−1−nm= (n−1)(1 +nm) +¹₂nm(m+ 1).

In a similar way, g_1,u1,...,u^k=

χ∈g_1,u1,...,u^k−1: λk=λⁱ_kα= 0 , so that dimg_1,u1,...,uk= dimg_1,...,uk−1−1−nm= (n−k)(1 +nm) +¹₂nm(m+ 1), and dimg1,u¹,...,uⁿ= ¹₂nm(m+ 1).

Repeating the same operation with g_1,u1,...,uⁿ,v¹ =

χ∈g_1,u1,...,uⁿ: µ₁=µⁱ_1α= 0 ,

we notice that all theµβ vanish automatically as everyλⁱ_jα vanishes ing_1,u1,...,uⁿ. Hence, we must only add the condition that the mn coefficients µⁱ_βα vanish for β = 1:

dimg_1,u1,...,uⁿ,v¹ = dimg_1,u1,...,uⁿ−nm=¹₂nm(m+ 1)−nm= ¹₂nm(m−1).

In an analogous way, when passing from g_1,u1,...,uⁿ,v¹,...,v^γ−1 to g_1,u1,...,uⁿ,v¹,...,v^γ

we must eliminate theµⁱ_γαwithα≥γ. Hence

dimg1,u¹,...,uⁿ,v¹,...,v^γ = dimg_1,u1,...,uⁿ,v¹,...,v^γ−1−n(m−γ+ 1)

= ¹₂n(m−(γ−2))(m−(γ−1))−n(m−γ+ 1)

= ¹₂n(m−γ+ 1)(m−γ), and so dimg1,u¹,...,uⁿ,v¹,...,v^m = 0.

We have still to calculate the dimension ofg2= kerσ2(Φ), where σ₂=σ₂(Φ) : S²T^∗M⊗ ∧ⁿ₂T^∗M →T^∗M ⊗ ∧ⁿ⁺¹₃ T^∗M,

is the prolongation of the symbol. It is defined as follows: Let [f] be the equivalence class of a functionf ∈C^∞(M),modulo sums with functions with vanishing second- order derivatives. The prolongation of the symbol is obtained by applying

S²T^∗M⊗ ∧ⁿ₂T^∗M →J¹(∧ⁿ⁺¹₃ T^∗M) [f]_y⊗ζ_y7→j¹_y(df∧ζ),

and then restricting to the associated vector bundle ofJ¹(∧ⁿ⁺¹₃ T^∗M).Hence, the expression ofσ₂ in local coordinates is the following (wherestands for the sym- metric product):

σ2(dx^jdx^k⊗v) = 0,

σ2(dx^jdy^β⊗v) =dx^j⊗(dy^β∧v),

σ2(dy^βdy^γ⊗v) =dy^β⊗(dy^γ∧v) +dy^γ⊗(dy^β∧v), σ2(dx^jdx^k⊗(dy^α∧vi)) = (−1)ⁱ

δ_i^kdx^j⊗(dy^α∧v) +δ_i^jdx^k⊗(dy^α∧v) σ2(dx^jdy^β⊗(dy^α∧vi)) =dx^j⊗(dy^β∧dy^α∧vi) + (−1)ⁱδ_i^jdy^β⊗(dy^α∧v), σ₂(dy^βdy^γ⊗(dy^α∧v_i)) =dy^β⊗(dy^γ∧dy^α∧v_i) +dy^γ⊗(dy^β∧dy^α∧v_i).

To calculate dimg2 we notice that an element

¯

χ= ¯λ_(jk)dx^jdx^k⊗v+ ¯λⁱ_(jκ)αdx^jdx^k⊗(dy^α∧v_i) + ¯µ_jβdx^jdy^β⊗v+ ¯µⁱ_jβαdx^jdy^β⊗(dy^α∧v_i) + ¯ν_(βγ)dy^βdy^γ⊗v+ ¯ν_(βγ)αⁱ dy^βdy^γ⊗(dy^α∧vi)

(where (nm) = (nm) are symmetric subindices) belongs tog2if and only if

¯

µjβ+X

i

(−1)ⁱ¯λⁱ_(ij)= 0

¯

ν(βγ)+X

i

(−1)ⁱµ¯ⁱ_iαβ = 0

¯

µⁱ_jβα+ ¯µⁱ_jαβ = 0

¯

ν_(βγ)αⁱ −ν¯_(αβ)γⁱ = 0

¯

ν_(βγ)αⁱ −ν¯_(αγ)βⁱ = 0.

Hence the dimension ofg2 can be counted as follows: Then(n+ 1)/2 coefficients λ¯_(jk) and the ¹₂n²(n+ 1)m coefficients ¯λⁱ_(,jk)α can be chosen freely. The latter completely determine the coefficients ¯µ_jβ. The n²m(m+ 1)/2 coefficients ¯µⁱ_jβα withα≥β can be freely chosen, and they determine the remaining ¯µⁱ_jβαand thus also the ¯ν(βγ) Finally, thenm(m+ 1)(m+ 2)/6 coefficients ¯ν_(βγ)αⁱ withα≥β ≥γ determine all the remaining ¯ν_(βγ)αⁱ . Hence,

dimg2=¹₂(nm+ 1)n(n+ 1) + ¹₂n²m(m+ 1) +¹₆nm(m+ 1)(m+ 2).

For the basis to be quasiregular (and hence, for the system to be involutive), the following dimension equality must hold:

(9) dimg2= dimg1+

n

X

j=1

dimg1,u¹,...,u^j +

m−1

X

β=1

dimg_1,u1,...,uⁿ,v¹,...,v^β. Moreover, we have

dimg1+

n−1

X

j=1

dimg_1,u1,...,u^j =

n−1

X

j=0

(n−j)(1 +nm) +¹₂nm(m+ 1)

= ¹₂n²m(m+ 1) +¹₂(1 +mn)n(n+ 1), and

dimg_1,u1,...,uⁿ+

m−1

X

γ=1

dimg_1,u1,...,uⁿ,v¹,...,v^γ =

m−1

X

γ=0 1

2n(m−γ+ 1)(m−γ)

=¹₆nm(m+ 1)(m+ 2),

as it can be easily shown by induction onm. Hence, the condition (9) is satisfied, and the differential system in question is involutive. (Lemma)

Now, all the obstructions for integrability must lie in the first prolongation.

The first prolongation of the system is

∂Fα

∂x^j = ∂²A⁰

∂x^j∂y^α − ∂²Aⁱ_α

∂xⁱ∂x^j, (10)

∂Fα

∂y^γ = ∂²A⁰

∂y^α∂y^γ − ∂²Aⁱ_α

∂xⁱ∂y^γ, (11)

∂F_αβⁱ

∂x^j = ∂²Aⁱ_α

∂x^j∂y^β − ∂²Aⁱ_β

∂x^j∂y^α, (12)

∂F_αβⁱ

∂y^γ = ∂²Aⁱ_α

∂y^β∂y^γ − ∂²Aⁱ_β

∂y^α∂y^γ. (13)

Let us thus look for every possible integrability condition, by checking all the linear relations that can be satisfied by the equations (6) (7), (10), (11), (12) and (13). The equations (6) and (10) cannot be related to any other equation because∂A⁰/∂y^αonly appears in theα-th component of (6) and, in a similar way,

∂²A⁰/∂x^j∂y^α only appears only in the (j, α)-th component of (10). Equations (7) can be related among themselves in order to obtain

(14) F_αβⁱ =−F_βαⁱ .

In an analogous way, equations (13) can be combined among themselves to give the conditions

∂F_αβⁱ

∂y^γ =−∂F_βαⁱ

∂y^γ ,

(which is a direct consequence of (14)), and also the conditions

(15) ∂F_αβⁱ

∂y^γ +∂F_βγⁱ

∂y^α +∂F_γαⁱ

∂y^β = 0.

Finally, the only possible way to eliminate theA’s in the equations (11) and (12) produce the condition

(16) ∂Fβ

∂y^α −∂Fα

∂y^β =∂F_αβⁱ

∂xⁱ .

Since we realise that this compatibility conditions are precisely the conditions (4) assuring that ξ is closed, we have obtained the formal integrability of the system of equations (6,7).

Finally, by using the Cartan-K¨ahler Theorem, we can assure the local integra- bility and hence the existence of a differentialn-form ω with the local expression (5) such that dω =ξ. Furthermore ω is the Poincar´e–Cartan form associated to the Lagrangian L = A⁰+Aⁱ_αy^α_i, whose Euler–Lagrange equations are precisely Fα=F_αβⁱ (∂y^β/∂xⁱ). Thus, the proof is complete.

Remark 5. In fact, we can drop the analiticity hypothesis, and obtain Theorem 2 for the case in which the equations (2) are justC^∞.The reason for that is that the equation R1 is associated to a differential operator with constant coefficients (as can be seen in (8)). Hence the Ehrenpreis–Malgrange Theorem (see [1, Chapter

X,1.2], and the references therein) can be used to state that formal integrability assures integrability in the C^∞ case.

References

[1] R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, P.A. Griffiths,Exterior Differential Systems, Springer-Verlag, New York, 1991.

[2] H. Kamo, R. Sugano,Necessary and Sufficient Conditions for the Existence of a Lagrangian.

The Case of Quasi-linear Field Equations, Ann. Physics128(1980), 298–313.

[3] V. Obˇadeanu, D. Opris,Le probl`eme inverse en Biodynamique, Seminarul de Mecanicˇa, Uni- versitatea din Timisoara33(1991).

Laboratoire Emile Picard C.N.R.S., U.M.R. 5580, Universit´e Paul Sabatier, UFR MIG, 118 Route de Narbonne, 31062 Toulouse, Cedex (France)

E-mail address:[email protected]

Instituto de F´isica Aplicada, CSIC , C/ Serrano 144, 28006-Madrid, Spain E-mail address:[email protected]

Departamento de Geometr´ia y Topolog´ia, Universidad Complutense de Madrid, Ciu- dad Universitaria s/n, 28040-Madrid, Spain

E-mail address:[email protected]