A GEOMETRIC HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES. Contents

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1 1 A GEOMETRIC HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES MANUEL DE LEÓN, JUAN CARLOS MARRERO, AND DAVID MARTÍN 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 Multisymplectic bundles Ehresmann Connections in the fibration π 1 : J 1 π M Hamiltonian sections The field equations 5 4. The Hamilton-Jacobi theory 6 5. Time-dependent mechanics 9 References Introduction The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q A ) (called the principal function) such that S t + H(qA, S qa) = 0. (1.1) 1 To 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 , MTM , project Ingenio Mathematica (i-math) No. CSD (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. 1

2 2 M. DE LEÓN, J.C. MARRERO, AND D. MARTÍN DE DIEGO If we put S(t, q A ) = W(q A ) te, where E is a constant, then W satisfies H(q A, W A) = E; (1.2) q 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 H = H(x µ, y i, p µ i ), where (xµ ) are coordinates in the space-time, (y i ) represent the field coordinates, and (p µ i ) are the conjugate momenta. In this context, the Hamilton-Jacobi equation is [17] S µ x µ + H(xν, y i, Sµ y i ) = 0 (1.3) where S µ = S µ (x ν, y j ). In this paper we introduce a geometric version for the Hamilton- Jacobi theory based in two facts: (1) the recent geometric description 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 ω Q = dq A dp A where (q A ) are coordinates in Q and (q A, p A ) are the induced ones in T Q. Let H : T Q R a hamiltonian function and X H the corresponding hamiltonian vector field: i XH ω Q = dh The integral curves of X H, (q A (t), p A (t)), satisfy the Hamilton equations: dq A = H dp A, = H dt p A dt q A

3 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 dq A dt = H p A 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: X λ H = Tπ Q X H λ as we can see in the following diagram: T Q X H T(T Q) λ π Q Tπ Q Q X λ H TQ Notice that the following conditions are equivalent: (i) If σ : I Q satisfies the equation dq A = H dt p A then λ σ is a solution of the Hamilton equations; (i) If σ : I Q is an integral curve of XH λ, then λ σ is an integral curve of X H ; (i) X H and XH λ are λ-related, i.e. Tλ(X λ H) = X H 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: (i) X λ H and X H are λ-related; (ii) d(h λ) = The multisymplectic formalism 3.1. Multisymplectic bundles. The configuration manifold in Mechanics is substituted by a fibred manifold π : E M

4 4 M. DE LEÓN, 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 (x µ, y i ) such that η = dx 1 dx n. We will use the following useful notations: d n x = dx 1 dx n d n 1 x µ = i x µ dn x. Denote by V π = ker Tπ the vertical bundle of π, that is, their elements are the tangent vectors to E which are π-vertical. Denote by Π : Λ n E E the vector bundle of n-forms on E. The total space Λ n E is equipped with a canonical n-form Θ: Θ(α)(X 1,...,X n ) = α(e)(tπ(x 1 ),...,TΠ(X n )) where X 1,...,X n T α (Λ n E) and α is an n-form at e E. The (n + 1)-form Ω = dθ, is called the canonical multisymplectic form on Λ n E. Denote by Λ n re the bundle of r-semibasic n-forms on E, say Λ n r E = {α Λn E i v1 v r α = 0, whenever v 1,..., v r are π-vertical} Since Λ n re is a submanifold of Λ n E it is equipped with a multisymplectic form Ω r, which is just the restriction of Ω. Two bundles of semibasic forms play an special role: Λ n 1 E and Λn 2 E. The elements of these spaces have the following local expressions: Λ n 1E : p 0 d n x Λ n 2E : p 0 d n x + p µ i dyi d n 1 x µ. which permits to introduce local coordinates (x µ, y i, p 0 ) and (x µ, y i, p 0, p µ i ) in Λ n 1E and Λ n 2E, respectively. Since Λ n 1E is a vector subbundle of Λ n 2E over E, we can obtain the quotient vector space denoted by J 1 π which completes the following exact sequence of vector bundles: 0 Λ n 1E Λ n 2E J 1 π 0. We denote by π 1,0 : J 1 π E and π 1 : J 1 π M the induced fibrations.

5 HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES Ehresmann Connections in the fibration π 1 : J 1 π M. A connection (in the sense of Ehresmann) in π 1 is a horizontal subbundle H which is complementary to V π 1 ; namely, T(J 1 π ) = H V π 1 where V π 1 = ker Tπ 1 is the vertical bundle of π 1. Thus, we have: (i) there exists a (unique) horizontal lift of every tangent vector to M; (ii) in fibred coordinates (x µ, y i, p µ i ) on J1 π, then V π 1 = span { y i, p µ },H = span {H µ }, i where H µ is the horizontal lift of x µ. (iii) there is a horizontal projector h : TJ π H Hamiltonian sections. Consider a hamiltonian section h : J 1 π Λ n 2 E of the canonical projection µ : Λ n 2E J 1 π which in local coordinates read as h(x µ, y i, p µ i ) = (xµ, y i, H(x, y, p), p µ i ). Denote by Ω h = h Ω 2, where Ω 2 is the multisymplectic form on Λ n 2 E. The field equations can be written as follows: i h Ω h = (n 1) Ω h, (3.1) where h denotes the horizontal projection of an Ehresmann connection in the fibred manifold π 1 : J 1 π M. The local expressions of Ω 2 and Ω h are: Ω 2 = d(p 0 d n x + p µ i dyi d n 1 x µ ) Ω h = d( H d n x + p µ i dyi d n 1 x µ ) The field equations. Next, we go back to the Equation (3.1). The horizontal subspaces are locally spanned by the local vector fields H µ = h( x µ) = x µ + Γi µ y i + (Γ µ) ν j where Γ i µ and (Γ µ) ν j are the Christoffel components of the connection. Assume that τ is an integral section of h; this means that τ : M J 1 π is a local section of the canonical projection π 1 : J 1 π M such that Tτ(x)(T x M) = H τ(x), for all x M. If τ(x µ ) = (x µ, τ i (x), τ µ i (x)) then the above conditions becomes τ i x µ = H p µ i, τ µ i x µ = H y i p ν j,

6 6 M. DE LEÓN, 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 λ = λ 0 (x, y) d n x + λ µ i (x, y) dyi d n 1 x µ. Alternatively, we can see it as a section λ : E Λ n 2E, and then we have λ(x µ, y i ) = (x µ, y i, λ 0 (x, y), λ µ i (x, y)). A direct computation shows that ( ) λ0 dλ = y λµ i dy i d n x + λµ i i x µ y j dyj dy i d n 1 x µ. Therefore, dλ = 0 if and only if λ 0 y i λ µ i y j = λµ i x µ (4.1) = λµ j y i. (4.2) 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 : T e E T e E h e (X) = Tπ 1,0 h (µ λ)(e) ǫ(x) where ǫ(x) T (µ λ)(e) (J 1 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 µ = h( x µ) = x µ + Γi µ((µ λ)(x, y)) y i. 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.

7 HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 7 Before to begin with the proof, let us consider some preliminary results. We have that is (h µ λ)(x µ, y i ) = (x µ, y i, H(x µ, y i, λ µ i (x, y)), λµ i (x, y)), h µ λ = H(x µ, y i, λ µ i (x, y)) dn x + λ µ i dyi d n 1 x µ. Notice that h µ λ is again a 2-semibasic n-form on E. A direct computation shows that ( H d(h µ λ) = y + H λ ν ) j i p ν j y + λµ i dy i d n x i x µ + λµ i y j dyj dy i d n 1 x µ. Therefore, we have the following result. Lemma 4.2. Assume dλ = 0; then d(h µ λ) = 0 if and only if H y + H λ ν j i p ν j y + λµ i i x = 0. µ Proof of the Theorem (i) (ii) It should be remarked the meaning of (i). Assume that is an integral section of h; then σ(x µ ) = (x µ, σ i (x)) σ i x = H µ p µ i. (i) states that in the above conditions, (µ λ σ)(x µ ) = (x µ, σ i (x), σ ν j = λν j (σ(x))) is a solution of the Hamilton equations, that is, σ µ i x µ = λµ i x µ + λµ i y j σ j x µ = H y i.

8 8 M. DE LEÓN, J.C. MARRERO, AND D. MARTÍN DE DIEGO Assume (i). Then H y + H λ ν j i p ν j y + λµ i i x µ = H y + H λ ν i i p ν j y + λµ i, (since dλ = 0) j x µ = H y + σj λ ν i i x ν y + λµ i j x, µ = 0 (since (i)) which implies (ii) by Lemma 4.2. (ii) (i) Assume that d(h µ λ) = 0. (since the first Hamilton equation) Since h is a flat connection, we may consider an integral section σ of h. Suppose that σ(x µ ) = (x µ, σ i (x)). Then, we have that σ i x = H µ p µ. i Thus, σ µ j x µ = λµ j x µ + λµ j y i σ i x µ, = λµ j x + λµ i σ i, (since dλ = 0) µ y j x µ = λµ j x µ + λµ i y j H p µ i, (since the first Hamilton equation) = H, (since (ii)). yj Assume that λ = ds, where S is a 1-semibasic (n 1)-form, say Therefore, we have S = S µ d n 1 x µ λ 0 = Sµ x, µ λµ i = Sµ y i and the Hamilton-Jacobi equation has the form ( ) S µ y i x + µ H(xν, y i, Sµ y ) = 0. i The above equations mean that S µ x µ + H(xν, y i, Sµ y i ) = f(xµ )

9 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): S µ x + H(x ν, y i, Sµ µ y ) = 0. i 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. We have the following identification Λ 1 2E = T E and we have local coordinates (t, y i, p 0, p i ) and (t, y i, p i ) on T E and J 1 π, respectively. The hamiltonian section is given by and therefore we obtain h(t, y i, p i ) = (t, y i, H(t, y, p), p i ), Ω h = dh dt dp i dy i. 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 (Ω h, dt) is a cosymplectic structure on E, that is, Ω h and dt are closed forms and dt Ω n h = dt Ω h Ω h is a volume form, where dime = 2n + 1. The Reeb vector field R h of the structure (Ω h, dt) satisfies i Rh Ω h = 0, i Rh dt = 1. The integral curves of R h are just the solutions of the Hamilton equations for H. The relation with the multisymplectic approach is the following: or, equivalently, h = R h dt, h( t ) = R h. A closed 1-form λ on E is locally represented by λ = λ 0 dt + λ i dy i. Using λ we obtain a vector field on E: (R h ) λ = Tπ 1,0 R h µ λ such that the induced connection is h = (R h ) λ dt

10 10 M. DE LEÓN, J.C. MARRERO, AND D. MARTÍN DE DIEGO Therefore, we have the following result. Theorem 5.1. The following conditions are equivalent: (i) (R h ) λ and R h 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 λ = ds = S S dt + t y i dyi, then we obtain the Hamilton-Jacobi equation ( S y i t + H(t, yi, S ) y i) = 0. References [1] R. Abraham, J.E. Marsden: Foundations of Mechanics (2nd edition). Benjamin-Cumming, Reading, [2] V.I. Arnold: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, Berlin, [3] E. Binz, J. Sniatycki, H. Fischer: Geometry of classical fields, North-Holland Mathematics Studies, 154, North-Holland Publishing Co., Amsterdam, [4] F. Cantrijn, A. Ibort, M. de León: On the geometry of multisymplectic manifolds. J. Austral. Math. Soc. (Series A) 66 (1999), [5] F. Cantrijn, A. Ibort, M. de León: Hamiltonian structures on multisymplectic manifolds. Rend. Sem. Mat. Univ. Pol. Torino, 54, 3 (1996), [6] J.F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda, N. Román-Roy: Geometric Hamilton-Jacobi theory. Int. J. Geom. Meth. Mod. Phys. 3 (7) (2006), [7] M.J. Gotay, J. Isenberg, J.E. Marsden, R. Montgomery: Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory. arxiv:physics/ v2. [8] M. de León, D. Iglesias-Ponte, D. Martín de Diego: Hamilton-Jacobi Theory for Nonholonomic Mechanical Systems. To appear in J. Phys. A: Math. Theor [9] M. de León, J.C. Marrero, D. Marín: A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories. In: New Developments in Differential Geometry (Debrecen, 1994), Mat. Appl. 350, Kluwer, Dordrecht, 1996, [10] M. de León, J.C. Marrero, D. Marín: Ehresmann connections in Classical Field Theories. In: Differential Geometry and its Applications (Granada, 1994), Anales de Física, Monografías, 2, 1995, [11] M. de León, D. Martín de Diego, A. Santamaría-Merino: Tulczyjew s triples and lagrangian submanifolds in classical field theory. In: Applied Differential Geometry and Mechanics, W. Sarlet and F. Cantrijn (eds). Universiteit Gent, pp [12] M. de León, D. Martín de Diego, A. Santamaría-Merino: Symmetries in Field Theory. International Journal of Geometric Methods in Modern Physics 1 (5) (2004),

11 HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 11 [13] M. de León, P.R. Rodrigues: Generalized Classical Mechanics and Field Theory. North-Holland, Amsterdam, [14] J.C. Marrero, D. Sosa: The Hamilton-Jacobi equation on Lie Affgebroids. International Journal of Geometric Methods in Modern Physics 3 (3), (2006), [15] C. Paufler, H. Romer: De Donder-Weyl equations and multisymplectic geometry. XXXIII Symposium on Mathematical Physics (Torún, 2001). Rep. Math. Phys. 49 (2002), [16] C. Paufler, H. Romer: Geometry of Hamiltonean n-vector fields in multisymplectic field theory. J. Geom. Phys. 44 (2002), [17] H. Rund: The Hamilton-Jacobi theory in the Calculus of Variations. Robert E. Krieger Publ. Co., Nuntington, N.Y [18] D.J. Saunders: The geometry of jet bundles, London Mathematical Society Lecture Notes Ser. 142, Cambridge Univ. Press, Cambridge, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Serrano 123, Madrid, Spain address: mdeleon@imaff.cfmac.csic.es J.C. Marrero: Departamento de Matemática Fundamental, Universidad de La Laguna, La Laguna, Canary Islands, Spain address: jcmarrer@ull.es D. Martín de Diego: Instituto de Ciencias Matemáticas, CSIC-UAM- UC3M-UCM, Serrano 123, Madrid, Spain address: d.martin@imaff.cfmac.csic.es