JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 1 JANUARY 2004 Lagrangian–Hamiltonian unified formalism for field theory Arturo Echeverrı´a-Enrı´quez Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain Carlos Lo´peza) Departamento de Matema´ticas, Campus Universitario. Fac. Ciencias, 28871 Alcala´ de Henares, Spain Jesu´s Marı´n-Solanob) Departamento de Matema´tica Econo´mica, Financiera y Actuarial, UB Avenida Diagonal 690, E-08034 Barcelona, Spain Miguel C. Mun˜oz-Lecandac) and Narciso Roma´n-Royd) Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain ͑Received 27 November 2002; accepted 15 September 2003͒ The Rusk–Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems ͑including dynamical equations and solutions, constraints, Legendre map, evolution operators, equiva- lence, etc.͒. In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1628384͔ I. INTRODUCTION In ordinary autonomous classical theories in mechanics there is a unified formulation of Lagrangian and Hamiltonian formalisms,1 which is based on the use of the Whitney sum of the ϭ ϵ ϫ ͑ tangent and cotangent bundles W TQ T*Q TQ QT*Q the velocity and momentum phase spaces of the system͒. In this space, velocities and momenta are independent coordinates. There is a canonical presymplectic form ⍀͑the pull-back of the canonical form in T*Q), and a natural i coupling function, locally expressed as piv , is defined by contraction between vectors and cov- ෈ ϱ ϭ i ectors. Given a Lagrangian L C (TQ), a Hamiltonian function, locally given by H piv ϪL(q,v), is determined, and, using the usual constraint algorithm for the geometric equation i(X)⍀ϭdH associated to the Hamiltonian system (W,⍀,H), we obtain that iϭ ץ ץ ͒ ͑ 1 The first constraint submanifold W1 is isomorphic to TQ, and the momenta L/ v pi are determined as constraints. ͑2͒ The geometric equation contains the second order condition viϭdqi/dt. ͑ ͒ ϵ 3 The identification W1 TQ allows us to recover the Lagrangian formalism. ͑4͒ The projection to the cotangent bundle generates the Hamiltonian formalism, including con- straints. The Legendre map and the time evolution operator are straightforwardly obtained by the previous identification and projection.2 a͒Electronic mail: [email protected] b͒Electronic mail: [email protected] c͒Electronic mail: [email protected] d͒Electronic mail: [email protected] 0022-2488/2004/45(1)/360/21/$22.00360 © 2004 American Institute of Physics Downloaded 21 Jul 2010 to 161.116.168.227. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp J. Math. Phys., Vol. 45, No. 1, January 2004 Lagrangian–Hamiltonian formalism for field theory 361 It is also worth noticing that this space is also appropriate for the formulation of different kinds of problems in optimal control.3–7 Furthermore, in Refs. 8 and 9 this unified formalism has been extended for nonautonomous mechanical systems. Our aim in this paper is to reproduce the same construction for first-order field theories, generating a unified description of Lagrangian and Hamiltonian formalisms and its correspon- dence, starting from the multisymplectic description of such theories. ͑See, for instance, Refs. 10–18, for some general references on this formalism. See, also, Refs. 19–25, for other geometric formulations of field theories.͒ As is shown throughout the paper, characteristics analogous to those pointed out for mechanical systems can be stated in this context. In Ref. 9, a first approach to this subject has been made, focusing mainly on the constraint algorithm for the singular case. The organization of the paper is as follows: Sec. II is devoted to reviewing the main features of the multisymplectic description of Lagrangian and Hamiltonian field theories. In Sec. III we develop the unified formalism for field theories: starting from the extended jet-multimomentum bundle ͑analogous to the Whitney sum in mechanics͒, we introduce the so-called extended Hamil- tonian system and state the field equations for sections, m-vector fields, connections, and jet fields in this framework. It is also shown how the standard Lagrangian and Hamiltonian descriptions are recovered from this unified picture. As a typical example, the minimal surface problem is de- scribed in this formalism in Sec. IV. Finally, we include an Appendix where basic features about connections, jet fields, and m-vector fields are displayed. Throughout this paper ␲:E!M will be a fiber bundle (dim Mϭm, dim EϭNϩm), where M is an oriented manifold with volume form ␻෈⍀m(M). ␲1:J1E!E is the jet bundle of local 1 1 1 ␣ A A sections of ␲, and ␲¯ ϭ␲ؠ␲ :J E!M gives another fiber bundle structure. (x ,y ,v␣) will denote natural local systems of coordinates in J1E, adapted to the bundle E!M (␣ϭ1,...,m; A ϭ ␻ϭ 1∧ ∧ mϵ m 1,...,N), and such that dx ••• dx d x. Manifolds are real, paracompact, connected, and Cϱ. Maps are Cϱ. Sum over crossed repeated indices is understood. II. GEOMETRIC FRAMEWORK FOR CLASSICAL FIELD THEORIES A. Lagrangian formalism ͑For details concerning the contents of this and the next section, see, for instance, Refs. 10–13, 17, 18, and 26–31. See, also, the Appendix͒. A classical field theory is described by giving a configuration fiber bundle ␲:E!M and a Lagrangian density, which is a ␲¯ 1-semibasic m-form on J1E usually written as LϭL␲¯ 1*␻, where L෈Cϱ(J1E) is the Lagrangian function determined by L and ␻. The Poincare´–Cartan m and (mϩ1)-forms associated with the Lagrangian density L are defined using the vertical endo- morphism V of the bundle J1E ͑see Ref. 30͒ ⌰ ͑V͒LϩL෈⍀m͑ 1 ͒ ⍀ Ϫ ⌰ ෈⍀mϩ1͑ 1 ͒ Lªi J E ; Lª d L J E . 1 A Lagrangian system is a couple (J E,⍀L). It is regular if ⍀L is a multisymplectic (mϩ1)-form ͑a closed m-form, mϾ1, is called multisymplectic if it is one-nondegenerate; elsewhere it is pre-multisymplectic͒. In natural charts in J1E we have ץ ץ Vϭ͑ AϪ A ␣͒ dy v␣dx A ␯ , xץ v␯ץ and Lץ Lץ A mϪ1 A m ⌰Lϭ dy ∧d x␮Ϫͩ v ϪL ͪ d x, A A ␮ v␮ץ v␮ץ Downloaded 21 Jul 2010 to 161.116.168.227. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp 362 J. Math. Phys., Vol. 45, No. 1, January 2004 Echeverrı´a-Enrı´quez et al. 2Lץ 2Lץ 2Lץ B A mϪ1 B A mϪ1 A B m ⍀LϭϪ dv␯ ∧dy ∧d x␣Ϫ dy ∧dy ∧d x␣ϩ v␣dv␯ ∧d x B A B A B A ␣vץ v␯ץ ␣vץ yץ ␣vץ v␯ץ 2Lץ Lץ 2Lץ ϩͩ vAϪ ϩ ͪ dy B∧dmx ␣ B A B ␣ B ␣vץ xץ yץ ␣vץ yץ mϪ1 ␣ m 2 A B v␯(¯y))Þ0, forץ␣vץ/L ץ)x )d x); the regularity condition is equivalent to detץ/ץ)where d x␣ϵi͑ every ¯y෈J1E. 1 The Lagrangian problem associated with a Lagrangian system (J E,⍀L) consists in finding sections ␾෈⌫(M,E), the set of sections of ␲, which are characterized by the condition 1␾͒ ͒⍀ ϭ ෈ 1 ͒ ͑ j *i͑X L 0, for every X X͑J E . In natural coordinates, if ␾(x)ϭ(x␣,␾A(x)), this condition is equivalent to demanding that ␾ satisfy the Euler–Lagrange equations Lץ ץ Lץ ͯ Ϫ ͩ ͪͯ ϭ0 ͑for Aϭ1,...,N͒. ͑1͒ A ␣ A ␣vץ xץ yץ j1␾ j1␾ The problem of finding these sections can be formulated equivalently as follows: finding a distribution D of T(J1E) such that it is integrable ͑that is, involutive͒, m-dimensional, ␲¯ 1-transverse, and the integral manifolds of D are the image of sections solution of the above equations ͑therefore, lifting of ␲-sections͒. This is equivalent to stating that the sections solution to the Lagrangian problem are the integral sections of one of the following equivalent elements: m 1 • A class of holonomic m-vector fields ͕XL͖ʚX (J E), such that i(XL)⍀Lϭ0, for every XL෈͕XL͖. 1 1 . A holonomic connection ٌL in ␲¯ :J E!M such that i(ٌL)⍀Lϭ(mϪ1)⍀L • 1 1 1 • A holonomic jet field ⌿L :J E!J J E, such that i(⌿L)⍀Lϭ0 ͑the contraction of jet fields with differential forms is defined in Ref. 11͒. Semi-holonomic locally decomposable m-vector fields, jet fields, and connections which are so- lution to these equations are called Euler–Lagrange m-vector fields, jet fields, and connections for 1 1 (J E,⍀L). In a natural chart in J E, the local expressions of these elements are ץ ץ ץ m A A XLϭ f ∧ ͩ ϩF ϩG ͪ , ␣ ␣␯ ␣ A A v␯ץ yץ xץ ϭ1␣ ץ ץ ץ ␣ A A , ͪ Lϭdx ͩ ϩF ϩGٌ ␣ ␣␯ ␣ A A v␯ץ yץ xץ ␣ A A A A ⌿Lϭ͑x ,y ,v␣ ,F␣ ,G␣␩͒, A A with F␣ϭv␣ ͑which is the local expression of the semi-holonomy condition͒, and where the A coefficients G␣␯ are related by the system of linear equations 2ץ 2ץ ץ 2ץ L A L L L A G␣␯ϭ Ϫ Ϫ v␯ ͑A,Bϭ1,...,N͒.