View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Digital.CSIC 1 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 ge- ometry 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. MART´IN 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 formal- ism [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 correspond- ing hamiltonian vector field: iXH ωQ = dH A The integral curves of XH ,(q (t),pA(t)), satisfy the Hamilton equa- tions: 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. MART´IN 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 ele- ments 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 multisym- plectic 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 sub- bundle 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. MART´IN 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)) .