A Geometric Hamilton-Jacobi Theory for Classical Field Theories

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A GEOMETRIC HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES

MANUEL DE LEON,´ JUAN CARLOS MARRERO, AND DAVID MART´IN DE DIEGO

Abstract. In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical ﬁeld theories in the framework of multisymplectic geometry and Ehresmann connections.

Contents

1. Introduction 1 2. A geometric Hamilton-Jacobi theory for Hamiltonian mechanics 2 3. The multisymplectic formalism 3 3.1. Multisymplectic bundles 3 1 ∗ 3.2. Ehresmann Connections in the ﬁbration π1 : J π −→ M 5 3.3. Hamiltonian sections 5 3.4. The ﬁeld equations 5 4. The Hamilton-Jacobi theory 6 5. Time-dependent mechanics 9 References 10

1. Introduction The standard formulation of the Hamilton-Jacobi problem is to ﬁnd a function S(t, qA) (called the principal function) such that ∂S ∂S + H(qA, ) = 0. (1.1) ∂t ∂qA

1To Prof. Demeter Krupka in his 65th birthday 2000 Mathematics Subject Classification. 70S05, 49L99. Key words and phrases. Multisymplectic field theory, Hamilton-Jacobi equations. This work has been partially supported by MEC (Spain) Grants MTM 2006- 03322, MTM 2007-62478, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. 1 2 M.DELEON,´ J.C. MARRERO, AND D. MARTÍN DE DIEGO

If we put S(t, qA) = W (qA) − tE, where E is a constant, then W satisfies ∂W H(qA, )= E; (1.2) ∂qA W is called the characteristic function. Equations (1.1) and (1.2) are indistinctly referred as the Hamilton- Jacobi equation. There are some recent attempts to extend this theory for classical field theories in the framework of the so-called multisymplectic formalism [15, 16]. For a classical field theory the hamiltonian is a function µ i µ µ i H = H(x ,y ,pi ), where (x ) are coordinates in the space-time, (y ) µ represent the field coordinates, and (pi ) are the conjugate momenta. In this context, the Hamilton-Jacobi equation is [17] ∂Sµ ∂Sµ + H(xν,yi, )=0 (1.3) ∂xµ ∂yi where Sµ = Sµ(xν,yj). In this paper we introduce a geometric version for the Hamilton- Jacobi theory based in two facts: (1) the recent geometric descrip- tion for Hamiltonian mechanics developed in [6] (see [8] for the case of nonholonomic mechanics); (2) the multisymplectic formalism for classical field theories [3, 4, 5, 7] in terms of Ehresmann connections [9, 10, 11, 12]. We shall also adopt the convention that a repeated index implies summation over the range of the index.

2. A geometric Hamilton-Jacobi theory for Hamiltonian mechanics

First of all, we give a geometric version of the standard Hamilton- Jacobi theory which will be useful in the sequel. Let Q be the conﬁguration manifold, and T ∗Q its cotangent bundle equipped with the canonical symplectic form A ωQ = dq ∧ dpA A A where (q ) are coordinates in Q and (q ,pA) are the induced ones in T ∗Q. ∗ Let H : T Q −→ R a hamiltonian function and XH the corresponding hamiltonian vector ﬁeld:

iXH ωQ = dH A The integral curves of XH ,(q (t),pA(t)), satisfy the Hamilton equations: A dq ∂H dpA ∂H = , = − A dt ∂pA dt ∂q HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 3

Theorem 2.1 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q (that is, dλ = 0 and, locally λ = dW ). Then, the following conditions are equivalent: (i) If σ : I → Q satisﬁes the equation dqA ∂H = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; (ii) d(H ◦ λ)=0.

To go further in this analysis, deﬁne a vector ﬁeld on Q: λ XH = T πQ ◦ XH ◦ λ as we can see in the following diagram:

∗ XH T Q / T (T ∗Q) ?

λ πQ TπQ

λ XH Q / T Q Notice that the following conditions are equivalent: (i) If σ : I → Q satisﬁes the equation dqA ∂H = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; λ (i)’ If σ : I → Q is an integral curve of XH , then λ◦σ is an integral curve of XH ; λ (i)” XH and XH are λ-related, i.e. λ Tλ(XH )= XH ◦ λ so that the above theorem can be stated as follows: Theorem 2.2 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q. Then, the following conditions are equivalent: λ (i) XH and XH are λ-related; (ii) d(H ◦ λ)=0.

3. The multisymplectic formalism 3.1. Multisymplectic bundles. The configuration manifold in Me- chanics is substituted by a fibred manifold π : E −→ M 4 M.DELEON,´ J.C. MARRERO, AND D. MARTÍN DE DIEGO such that (i) dim M = n, dim E = n + m (ii) M is endowed with a volume form η. We can choose fibred coordinates (xµ,yi) such that η = dx1 ∧···∧ dxn .

We will use the following useful notations: dnx = dx1 ∧···∧ dxn n−1 µ n d x = i ∂ d x . ∂xµ Denote by V π = ker T π the vertical bundle of π, that is, their elements are the tangent vectors to E which are π-vertical. Denote by Π:ΛnE −→ E the vector bundle of n-forms on E. The total space ΛnE is equipped with a canonical n-form Θ:

Θ(α)(X1,...,Xn)= α(e)(T Π(X1),...,T Π(Xn)) n where X1,...,Xn ∈ Tα(Λ E) and α is an n-form at e ∈ E. The (n + 1)-form Ω= −dΘ , is called the canonical multisymplectic form on ΛnE. n Denote by Λr E the bundle of r-semibasic n-forms on E, say n n Λr E = {α ∈ Λ E | iv1∧···∧vr α =0, whenever v1,...,vr are π-vertical} n n Since Λr E is a submanifold of Λ E it is equipped with a multisymplectic form Ωr, which is just the restriction of Ω. n n Two bundles of semibasic forms play an special role: Λ1 E and Λ2 E. The elements of these spaces have the following local expressions: n n Λ1 E : p0 d x n n µ i n−1 µ Λ2 E : p0 d x + pi dy ∧ d x . µ i µ i µ which permits to introduce local coordinates (x ,y ,p0)and(x ,y ,p0,pi ) n n in Λ1 E and Λ2 E, respectively. n n Since Λ1 E is a vector subbundle of Λ2 E over E, we can obtain the quotient vector space denoted by J 1π∗ which completes the following exact sequence of vector bundles: n n 1 ∗ 0 −→ Λ1 E −→ Λ2 E −→ J π −→ 0 . 1 ∗ 1 ∗ We denote by π1,0 : J π −→ E and π1 : J π −→ M the induced ﬁbrations. HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 5

1 ∗ 3.2. Ehresmann Connections in the ﬁbration π1 : J π −→ M. A connection (in the sense of Ehresmann) in π1 is a horizontal subbundle H which is complementary to V π1; namely, 1 ∗ T (J π )= H ⊕ V π1 where V π1 = ker T π1 is the vertical bundle of π1. Thus, we have: (i) there exists a (unique) horizontal lift of every tangent vector to M; µ i µ 1 ∗ (ii) in ﬁbred coordinates (x ,y ,pi ) on J π , then ∂ ∂ H H V π1 = span { i , µ } , = span { µ} , ∂y ∂pi ∂ where Hµ is the horizontal lift of ∂xµ . (iii) there is a horizontal projector h : TJ ∗π −→ H.

3.3. Hamiltonian sections. Consider a hamiltonian section 1 ∗ n h : J π −→ Λ2 E n 1 ∗ of the canonical projection µ :Λ2 E −→ J π which in local coordinates read as µ i µ µ i µ h(x ,y ,pi )=(x ,y , −H(x,y,p),pi ) . ∗ n Denote by Ωh = h Ω2, where Ω2 is the multisymplectic form on Λ2 E. The ﬁeld equations can be written as follows:

ih Ωh =(n − 1) Ωh , (3.1) where h denotes the horizontal projection of an Ehresmann connection 1 ∗ in the ﬁbred manifold π1 : J π −→ M.

The local expressions of Ω2 and Ωh are: n µ i n−1 µ Ω2 = −d(p0 d x + pi dy ∧ d x ) n µ i n−1 µ Ωh = −d(−Hd x + pi dy ∧ d x ) . 3.4. The field equations. Next, we go back to the Equation (3.1). The horizontal subspaces are locally spanned by the local vector fields ∂ ∂ ∂ ∂ h i ν Hµ = ( µ )= µ +Γµ i + (Γµ)j ν , ∂x ∂x ∂y ∂pj i ν where Γµ and (Γµ)j are the Christoffel components of the connection. Assume that τ is an integral section of h; this means that τ : M −→ 1 ∗ 1 ∗ J π is a local section of the canonical projection π1 : J π −→ M such that T τ(x)(TxM)= Hτ(x), for all x ∈ M. µ µ i µ If τ(x )=(x , τ (x), τi (x)) then the above conditions becomes i µ ∂τ ∂H ∂τi ∂H µ = µ , µ = − i ∂x ∂pi ∂x ∂y 6 M.DELEON,´ J.C. MARRERO, AND D. MARTÍN DE DIEGO which are the Hamilton equations.

4. The Hamilton-Jacobi theory Let λ be a 2-semibasic n-form on E; in local coordinates we have

n µ i n−1 µ λ = λ0(x, y) d x + λi (x, y) dy ∧ d x .

n Alternatively, we can see it as a section λ : E −→ Λ2 E, and then we have µ i µ i µ λ(x ,y )=(x ,y ,λ0(x, y),λi (x, y)) . A direct computation shows that ∂λ ∂λµ ∂λµ dλ = 0 − i dyi ∧ dnx + i dyj ∧ dyi ∧ dn−1xµ . ∂yi ∂xµ ∂yj Therefore, dλ = 0 if and only if ∂λ ∂λµ 0 = i (4.1) ∂yi ∂xµ ∂λµ ∂λµ i = j . (4.2) ∂yj ∂yi Using λ and h we construct an induced connection in the ﬁbred manifold π : E −→ M by deﬁning its horizontal projector as follows:

h˜e : TeE −→ TeE ˜ he(X) = T π1,0 ◦ h(µ◦λ)(e) ◦ ǫ(X)

1 ∗ where ǫ(X) ∈ T(µ◦λ)(e)(J π ) is an arbitrary tangent vector which projects onto X. From the above definition we immediately proves that (i) h˜ is a well-defined connection in the fibration π : E −→ M. (ii) The corresponding horizontal subspaces are locally spanned by ∂ ∂ ∂ H˜ = h˜( )= +Γi ((µ ◦ λ)(x, y)) . µ ∂xµ ∂xµ µ ∂yi The following theorem is the main result of this paper. Theorem 4.1. Assume that λ is a closed 2-semibasic form on E and that h˜ is a flat connection on π : E −→ M. Then the following conditions are equivalent: (i) If σ is an integral section of h˜ then µ ◦ λ ◦ σ is a solution of the Hamilton equations. (ii) The n-form h ◦ µ ◦ λ is closed. HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 7

Before to begin with the proof, let us consider some preliminary results. We have

µ i µ i µ i µ µ (h ◦ µ ◦ λ)(x ,y )=(x ,y , −H(x ,y ,λi (x, y)),λi (x, y)) , that is

µ i µ n µ i n−1 µ h ◦ µ ◦ λ = −H(x ,y ,λi (x, y)) d x + λi dy ∧ d x .

Notice that h ◦ µ ◦ λ is again a 2-semibasic n-form on E. A direct computation shows that

ν µ ∂H ∂H ∂λj ∂λi i n d(h ◦ µ ◦ λ) = − i + ν i + µ dy ∧ d x ∂y ∂pj ∂y ∂x ∂λµ + i dyj ∧ dyi ∧ dn−1xµ . ∂yj

Therefore, we have the following result. Lemma 4.2. Assume dλ =0; then d(h ◦ µ ◦ λ)=0 if and only if ν µ ∂H ∂H ∂λj ∂λi i + ν i + µ =0 . ∂y ∂pj ∂y ∂x

Proof of the Theorem (i) ⇒ (ii) It should be remarked the meaning of (i). Assume that σ(xµ)=(xµ, σi(x)) is an integral section of h˜; then ∂σi ∂H µ = µ . ∂x ∂pi

(i) states that in the above conditions,

µ µ i ν ν (µ ◦ λ ◦ σ)(x )=(x , σ (x), σ¯j = λj (σ(x))) is a solution of the Hamilton equations, that is, ∂σ¯µ ∂λµ ∂λµ ∂σj ∂H i = i + i = − . ∂xµ ∂xµ ∂yj ∂xµ ∂yi 8 M.DELEON,´ J.C. MARRERO, AND D. MART´IN DE DIEGO

Assume (i). Then ν µ ∂H ∂H ∂λj ∂λi i + ν i + µ ∂y ∂pj ∂y ∂x ν µ ∂H ∂H ∂λi ∂λi = i + ν j + µ , (since dλ = 0) ∂y ∂pj ∂y ∂x ∂H ∂σj ∂λν ∂λµ = + i + i , (since the first Hamilton equation) ∂yi ∂xν ∂yj ∂xµ = 0 (since (i)) which implies (ii) by Lemma 4.2. (ii) ⇒ (i) Assume that d(h ◦ µ ◦ λ)=0. Since h˜ is a flat connection, we may consider an integral section σ of h˜. Suppose that σ(xµ)=(xµ, σi(x)). Then, we have that ∂σi ∂H µ = µ . ∂x ∂pi Thus, ∂σ¯µ ∂λµ ∂λµ ∂σi j = j + j , ∂xµ ∂xµ ∂yi ∂xµ ∂λµ ∂λµ ∂σi = j + i , (since dλ = 0) ∂xµ ∂yj ∂xµ µ µ ∂λj ∂λi ∂H = µ + j µ , (since the first Hamilton equation) ∂x ∂y ∂pi ∂H = − , (since (ii)). ∂yj

Assume that λ = dS, where S is a 1-semibasic (n − 1)-form, say S = Sµ dn−1xµ Therefore, we have ∂Sµ ∂Sµ λ = , λµ = 0 ∂xµ i ∂yi and the Hamilton-Jacobi equation has the form ∂ ∂Sµ ∂Sµ + H(xν,yi, ) =0 . ∂yi ∂xµ ∂yi The above equations mean that ∂Sµ ∂Sµ + H(xν,yi, )= f(xµ) ∂xµ ∂yi HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 9 so that if we put H˜ = H − f we deduce the standard form of the Hamilton-Jacobi equation (since H and H˜ give the same Hamilton equations): ∂Sµ ∂Sµ + H˜ (xν,yi, )=0 . ∂xµ ∂yi An alternative geometric approach of the Hamilton-Jacobi theory for Classical Field Theories in a multisymplectic setting was discussed in [15, 16].

5. Time-dependent mechanics A hamiltonian time-dependent mechanical system corresponds to a classical field theory when the base is M = R. 1 ∗ We have the following identification Λ2E = T E and we have local i i ∗ 1 ∗ coordinates (t, y ,p0,pi) and (t, y ,pi) on T E and J π , respectively. The hamiltonian section is given by i i h(t, y ,pi)=(t, y , −H(t,y,p),pi) , and therefore we obtain i Ωh = dH ∧ dt − dpi ∧ dy . If we denote by η = dt the different pull-backs of dt to the fibred manifolds over M, we have the following result.

The pair (Ωh, dt) is a cosymplectic structure on E, that is, Ωh and n dt are closed forms and dt ∧ Ωh = dt ∧ Ωh ∧···∧ Ωh is a volume form, where dimE = 2n + 1. The Reeb vector ﬁeld Rh of the structure (Ωh, dt) satisﬁes

iRh Ωh =0 , iRh dt =1.

The integral curves of Rh are just the solutions of the Hamilton equations for H. The relation with the multisymplectic approach is the following:

h = Rh ⊗ dt , or, equivalently, ∂ h( )= R . ∂t h A closed 1-form λ on E is locally represented by i λ = λ0 dt + λi dy . Using λ we obtain a vector ﬁeld on E:

(Rh)λ = T π1,0 ◦ Rh ◦ µ ◦ λ such that the induced connection is

h˜ =(Rh)λ ⊗ dt 10 M. DE LEON,´ J.C. MARRERO, AND D. MART´IN DE DIEGO

Therefore, we have the following result. Theorem 5.1. The following conditions are equivalent:

(i) (Rh)λ and Rh are (µ ◦ λ)-related. (ii) The 1-form h ◦ µ ◦ λ is closed. Remark 5.2. An equivalent result to Theorem 5.1 was proved in [14] (see Corollary 5 in [14]). ⋄

Now, if ∂S ∂S λ = dS = dt + dyi , ∂t ∂yi then we obtain the Hamilton-Jacobi equation ∂ ∂S ∂S + H(t, yi, ) =0 . ∂yi ∂t ∂yi

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Instituto de Ciencias Matematicas,´ CSIC-UAM-UC3M-UCM, Serrano 123, 28006 Madrid, Spain E-mail address: [email protected]

J.C. Marrero: Departamento de Matematica´ Fundamental, Univer- sidad de La Laguna, La Laguna, Canary Islands, Spain E-mail address: [email protected]

D. Mart´ın de Diego: Instituto de Ciencias Matematicas,´ CSIC-UAM- UC3M-UCM, Serrano 123, 28006 Madrid, Spain E-mail address: [email protected]