Lagrangian and Hamiltonian formalism in Field Theory: a simple model

The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for the problem and describe the Lagrangian and Hamiltonian formalism. We outline also natural generalizations of this approach to arbitrary dimensions.


Introduction
The main purpose of this work is to implement the Tulczyjew triple approach of the Analytical Mechanics [26,27] into the statics of multi-dimensional objects, i.e. smooth maps from a disc D ⊂ R n into a manifold M . This problem can be regarded as a toy model for the Classical Field Theory, since the set of smooth maps from R n to M can be treated as a set of sections of the trivial bundle pr 1 : R n × M → R n . In comparison with general geometric approaches [3,4,25] the situation is considerably simplified, because the bundle is trivial and the base manifold R n has a canonical volume form and a canonical base of sections of the tangent bundle. For n = 1 and M being the space of configurations of a mechanical system we recover the model of the autonomous mechanics.
We work with this geometrically simple version of the Classical Field Theory to present the main ideas of our approach to the Lagrangian and Hamiltonian formalism that differs from the ones which are present in the literature [15]. Since we skipped topological difficulties in this case, we could concentrate on the recognition of physically important objects, like the phase space, phase dynamics, Legendre map, Hamiltonian, etc. These issues are usually not elaborated well in the literature, as the Classical Field Theory models use to concentrate on the Euler-Lagrange equations. Of course, we recover also the commonly accepted Euler-Lagrange equations, this time without requiring any regularity of the Lagrangian.
The methods we use are based on expressing the theory in terms of differential relations rather than maps or tensor fields. For the price of dealing with differential calculus of relations we get, in our opinion, better understanding of geometric structures involved. It was also shown in [10,8] that using the same philosophy one can pass easily to the more complicated geometrical framework based on Lie or general algebroids. In the case of Analytical Mechanics similar generalizations were proposed by many authors (e.g. [19,20,17], but the approach presented in [10,8], being ideologically simpler, will be our starting point.
We would like to point out that all the constructions we perform are motivated by the variational calculus that we consider to be the fundamental idea of Classical Mechanics and Field Theory. The origin of geometric structures we use lies in the rigorous formulation of the variational principle including boundary terms that one can find in [25,22]. Nevertheless, we do not enter into details of the variational calculus and we treat it rather as a guide-line for recognizing which geometrical structures are appropriate in this case.
The problem itself, i.e. the generalization of the symplectic framework for autonomous mechanics to higher dimensions is not new and was first treated by Günther in [12]. The underlying geometric structure of Günther's theory, known as k-symplectic structure, was described systematically in [1,2]. Recently, Rey, Roman-Roy, Salgado and Valarino renewed the theory and described its Lie algebroid version [24]. Our work is also related to the multisymplectic approach to the Classical Field Theory developed by Gotay, Isennberg, Marsden and others and presented e.g. in [3,4,5,6]. The Tuczyjew triple in the context of multisymplectic field theories appeared already in [18].
For the presentation of our general idea, let us first recall the description of the dynamics of a classical autonomous mechanical system without constraints. Let M denote the manifold of positions of the system. The trajectory is therefore a smooth path in M , i.e. a map from the time interval [t 0 , t 1 ] ⊂ R into M . We can try to describe our system in variational way, looking for those trajectories γ : R → M that, for the fixed time interval [t 0 , t 1 ], minimize the action functional L(tγ(t))dt.
We assumed above that the Lagrangian is of first-order i.e. it is a function on the tangent bundle TM . The curve t → tγ(t) will denote the tangent prolongation of the curve γ in M . The variational approach for the finite time interval leads to the Euler-Lagrange equations and the definition of momenta. The space of momenta is usually called the phase space of the system. In the case of autonomous mechanics, the phase space is just the cotangent bundle T * M . We describe the system by a first-order differential equation on the phase space, called the phase dynamics. The phase dynamics D is described by a subset of TT * M : where α M is the Tulczyjew isomorphism (defined in [26]) α M : TT * M → T * TM and dL(TM ) is the image of the differential of the Lagrangian. A curve t → η(t) ∈ T * M satisfies the phase dynamics if its tangent prolongation lies in D. A curve t → γ(t) ∈ TM satisfies, in turn, the corresponding Euler-Lagrange equation, if the curve t → α −1 M (dL(γ(t))) ∈ TT * M is the tangent prolongation of its projection to T * M (see [10,8]).
All the structures needed for generating the dynamics from the Lagrangian can be summarized in the following commutative diagram:  where β M is the canonical isomorphism between TT * M and T * T * M given by the canonical symplectic form ω M on T * M , Let us recall for the future reference that the canonical symplectic form ω M is defined by where ϑ M is the Liouville form given by The structures needed for Hamiltonian mechanics can be presented in the following commutative diagram: T * T * M

Variational approach
We start with Variational Calculus which is our guide-line for recognizing geometrical objects representing physical quantities. Let L be a smooth function on the manifold 2 TM of the first jets of maps from C ∞ (R 2 , M ); we will call L a Lagrangian. Any Lagrangian defines an action functional S on maps u : D → M from the unit disc D ⊂ R 2 into M: where u a (x) = q a (u(x)) and u a . Note that the fact that Lagrangian can be just a function on 2 T M is due to the existence of the canonical volume form dx 1 ∧ dx 2 on R 2 . We can therefore identify scalar densities, i.e. objects that can be integrated, with functions.
Variations of u are maps δu from D to TM covering u: is a vector tangent to the curve t → χ(t, x) at t = 0. In the following we perform the standard calculus of a variation of S with respect to the variation δu: Using the Stokes theorem, we obtain where the last integral is calculated over ∂D oriented as in the Stokes theorem, using the canonical orientation of R 2 . Looking for the stationary points of the action functional S we put the condition dS, δu = 0 for every δu, which means that Field Theory: a simple model on the disc and ∂L ∂q a on the boundary ∂D. The equation (3.1) is traditionally called the Euler-Lagrange equation. The boundary term is an analog of the momentum in the Classical Mechanics. The momentum evaluated on a variation δu gives a one-form on R 2 (to be integrated over ∂D). It follows that the phase space is a space of covectors on M with values in the cotangent space of R 2 . The cotangent bundle of R 2 is trivial, with the fiber being just (R 2 ) * ≃ R 2 , but we will keep the notation (R 2 ) * and use the basis (dx 2 , −dx 1 ) to identify it with R 2 . The phase space can be therefore identified with The Legendre map which associates a momentum to an infinitesimal configuration will be discussed later in the section.
In the calculation of the differential of the action functional we have used implicitly a mapping T M at t = 0. From the same homotopy we get The first jet at x of the last map is an element of 2 T TM . Therefore The above definition is analogous to the definition of the canonical flip κ M : TTM → TTM .
Using the local coordinate system (q a ) on M , we can construct local coordinates on T TM will be denoted by and the ones on T 2 T M by 2 , δq f 2 ), using the same notation for coordinates in different spaces does not lead to any confusion.

The Lagrangian side
In the previous section we recognized the phase space as 2 T * M . In the following it will be useful to remember that the space Both bundles are vector bundles which form together a double vector bundle [16]: T TM , respectively, such that they have the same projection on 2 T M . Let p and δu denote the representatives covering the same map u : R 2 → M .
Interpreting an element of 2 T * M as a covector on M with values in (R 2 ) * , we can define the mapping where the target space is the fiber of T * R 2 . The mapping can be viewed as a one-form on R 2 . The differential of the above one-form is a two-form on R 2 which, due to the existence of the canonical form dx 1 ∧ dx 2 , can be identified with a function. The formula On the other hand, in the space 2 TTM of jets of variations we have also an adapted set of coordinates induced by the coordinates on M : In the above coordinates the evaluation reads: Now we are ready to define the mapping α : The mapping α is an analog of α M : TT * M → T * TM used by Tulczyjew in the autonomous mechanics. The important difference with the case of Classical Mechanics is that α is no longer an isomorphism, therefore α −1 is a relation only, not a mapping. In local coordinates we obtain (4.6)ṗ 1 a 1 +ṗ 2 The Legendre map, that associates a momentum to an infinitesimal configuration, is defined as: In coordinates it reads The Euler-Lagrange equations for configurations can be formulated in the following way The equations we obtained are in full agreement with equations commonly accepted in Classical Filed Theory the theory (cf. [3,13]). All the structure needed for generating the phase equations from the Lagrangian can be presented in the following diagram: | | y y y y y y y y

The Hamiltonian side
In the autonomous mechanics the basic structure is the canonical symplectic form ω M on the cotangent bundle being the phase space. Using the form ω M , we associate the Hamiltonian vector field to any Hamiltonian -a smooth function on the phase space. In our case, the phase space is not a symplectic manifold any more, but still we can establish a correspondence between the cotangent bundle of the phase space and the space 2 T 2 T * M of jets of maps from C ∞ (R 2 , 2 T * M ). We present here two ways of constructing the appropriate mapping.
The first method is based on the fact that the phase bundle is a vector bundle over M , so we have the canonical antisymplectomorphism (cf. [7,16]) Denoting the above antisymplectomorphism by R and composing it with α we obtain Since R and α are double vector bundle morphisms, we obtain the following diagram for the mapping β: 3  3  3  3  3  3  3  3  3  3  3  3  3  3  3 3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3 and the system of coordinates (q a , p 1 b , p 2 c , ϕ d , ψ e 1 , ψ f 2 ), derived from the coordinates (q a , p 1 b , p 2 c ) and associated to the local sections (dq a , dp 1 b , dp 2 c ), we get that

For any Hamiltonian
H : the phase dynamic is represented by the subset Also in this case, β −1 is a relation only. In local coordinates we obtain the phase equations An alternative way of constructing the mapping β does not refer to the map α. Let us denote by pr 1 , pr 2 the projections on the first and the second factor of 2 T * M = TM × M TM . In local coordinates, we have Applying the tangent lift to the both projections we obtain Composing the cartesian product of the above tangent mappings with the inclusion ı : To Finally, we end up with the map Proposition 5.1. The mappings defined in (5.1) and (5.5) coincide, i.e.
Proof: Let us start with recalling the definition of the canonical isomorphism R E for a general vector bundle E → M . The graph of R E is the Lagrangian submanifold generated in We see that, by definition, for any element ϕ ∈ T * E, its image R E (ϕ) has the same projections onto E and E * as ϕ. If we take now two curves covering the same curve in M and such that γ(0) and η(0) are equal to the projections of ϕ to E and E * respectively, we can write (ϕ, R E (ϕ)), (tγ(0), tη(0)) = d dt |t=0 η(t), γ(t) , Let now ψ : R 2 → T * M be a homotopy such that ψ(0, 0) is the projection of v and w on T * M , the curve a → ψ(a, 0) is a representative of v, and b → ψ(0, b) is a representative of w. Using the definitions of ω M and ϑ M (see (1.5,1.6), we can write that We can simplify the above formula a little bit introducing curves which represent v and w, respectively, and a homotopy in M defined by ρ = π M • ψ.
In the new notation we have Since our model is a very simple and designed to study geometrical structures related to the Classical Field Theory rather than to describe real physical systems, it is not easy to find physically important examples. In the literature, one can find the so called bosonic string theory. There are two approaches to the subject. In one of them, due to Polyakow [23], configurations are mappings from a two-dimensional manifold X of the string into the product of Minkowski space and the space of symmetric tensors on X. It means that not only the space-time position of the string is subject to variations, but also the metric on the string itself. In the simpler approach by Nambu [11,21], the metric on the string is fixed to be the pull-back of the Minkowski metric by the string space-time configuration. In the Nambu approach we deal therefore with mappings from the two-dimensional manifold to the Minkowski space. In our example we will use the Nambu version with another simplification by taking X = R 2 . The Minkowski space (M, V, η) is a four-dimensional affine space with the model vector space V equipped with a bilinear symmetric form η of signature (+ − − −). We will denote byη the associated self-adjoint map from V to V * . Using the affine structure, we can identify the tangent bundle τ M : TM → M with the trivial bundle M ×V → M , and the cotangent bundle π M : T * M → M with the trivial bundle M × V * → M . The spaces that appear in the Lagrangian picture are therefore The first jet of a mapping u : R 2 → M at the point (x 1 , x 2 ) is identified with a triple (q, v 1 , v 2 ), where q = u(x 1 , x 2 ), v 1 is a vector tangent to the curve t → u(x 1 + t, x 2 ) at t = 0, and v 2 is a vector tangent to the curve t → u(x 1 , x 2 + t) at t = 0. The Lagrangian at the point j 1 u is the scalar density associated to u * η which (after identification with the function on M × V × V ) gives The The Legendre map is in our example reversible, therefore we can express infinitesimal configurations in terms of momenta: In the above formulae we used the same letter η for the bilinear form associated to η on the dual side. The matrix g, in terms of momenta, takes the form g = −η(p 2 , p 2 ) η(p 1 , p 2 ) η(p 1 , p 2 ) −η(p 1 , p 1 ) .

Conclusions
We have presented a toy model of a Classical Field Theory to introduce main concepts of a new approach to Lagrange and Hamilton formalisms. The starting point was the Tulczyjew triple in the Classical Mechanics, generalized now to the case of fields. In this approach all main ingredients are present: starting with a Lagrangian, not only the Euler-Lagrange field equation has been derived, but also the phase space and phase dynamics have been recognized, together with the Legendre map and the Hamiltonian picture. The latter suggests that momenta are dual rather to infinitesimal variations (displacements) than to infinitesimal configurations ('velocities'). The main difference with respect to the classical situation is that, to construct the phase dynamics, relations are used instead of mappings. This approach, presented here for maps from the disc into a manifold, can be naturally generalized to sections of a fibration and to an 'algebroid' setting as well. We postpone these studies to a separate paper.