Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T*TM, T*TM, and TT*M. We show that there exist an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures of degree n, i.e. objects generalizing vector bundles (for which n=1). For instance, the role of TT*M is played in our approach by the vector bundle of n-vectors on the bundle of n-covectors, which is canonically a graded bundle of degree n over the bundle of n-vectors. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.


Introduction
This work is an account of a research undertaken jointly with W. M. Tulczyjew on the Legendre transformation in the dynamics of strings. Let us recall structures which are involved in the Legendre transformation for the dynamics of point-like objects, i.e. when the motion is given by a one-dimensional submanifold (parametrized or not) in the configuration manifold M . An infinitesimal piece of motion is a tangent vector and a system is described by a Lagrangian, i.e. a function (constrained or not) on ÌM. The geometric content of Lagrangian and Hamiltonian formulation of the dynamics is contained in the following diagram. (1.1) The dynamics is a subset of ÌÌ * M , which can be transported either to Ì * ÌM (Lagrangian side) or to Ì * Ì * M (Hamiltonian side). The top elements in the diagram are certain canonical double vector bundles over ÌM and Ì * M . A double vector bundle is, roughly speaking, a manifold with two compatible vector bundle structures, i.e. two vector bundle structures whose Euler vector fields commute (see Section (4) For more details see [29], [32], [36], [45], and [47]. Similar constructions in the context of field theory one can find e.g. in [15,16,19], while the algebroid version is discussed in [17,18,20].
In the following, we shall give similar constructions, motivated by the study of dynamics of one dimensional, non parametrized objects, in which the bundle ∧ 2 ÌM of tangent bivectors replaces ÌM. The motion of a system is given by a two-dimensional submanifold in the manifold M ("space-time"). An infinitesimal piece of the motion is the jet of the submanifold. For first-order theories it is the first jet. We immediately realize that, for this choice of infinitesimal motions, the infinitesimal action (Lagrangian) is not a function on first jets, but a section of certain line bundle over first jets manifold, the dual bundle of the bundle of "first jets with volumes". This leads to essential complications even in one dimensional case (relativistic particle), see [34] and an alternative approach in [39].
A compromise is to take for the space of infinitesimal pieces of motions the space of simple 2-vectors. Non-zero 2-vectors represent first jets of 2-dimensional submanifolds (being 2-dimensional subspaces in tangent spaces) together with a volume which should be taken so that it takes the value 1 on the 2-vector. It is convenient to extend this space to all 2-vectors, i.e. to the vector bundle ∧ 2 ÌM. A Lagrangian is a positive homogeneous function on ∧ 2 ÌM (thus the action functional does not depend on the parametrization of the surface) and the corresponding Hamiltonian (if it exists) is a function on the dual vector bundle ∧ 2 Ì * M . The dynamics should be an equation (possibly implicit) for 2-dimensional submanifolds in the phase space, i.e. a subset D in ∧ 2 Ì ∧ 2 Ì * M . A surface S in the phase space ∧ 2 Ì * M is a solution for the phase dynamics if and only if its tangent space at ω ∈ ∧ 2 Ì * M is represented by a bivector from D ω . If we use a parametrization, then the tangent bivectors associated with this parametrization must belong to D.
We realize immediately, that some of the tools, suitable for point-like objects, here are not adequate. On one hand, Ì * ∧ 2 Ì * M and Ì * ∧ 2 ÌM are symplectic manifolds with double vector bundle structures, but it is not true for ∧ 2 Ì ∧ 2 Ì * M . In general, for a vector bundle τ : E → N , the fibration ∧ 2 Ìτ : ∧ 2 ÌE → ∧ 2 ÌN is not a vector bundle. Moreover, we do not have an identification of ∧ 2 Ì(E × N F ) with ∧ 2 ÌE × ∧ 2 TN ∧ 2 ÌF and the total derivative d T seems to be useless.
On the other hand, we have the canonical multisymplectic 3-form ω 2 M on ∧ 2 Ì * M , which gives, by the contraction, canonical morphism β 2 M : ∧ 2 Ì ∧ 2 Ì * M → Ì * ∧ 2 Ì * M and we can replace the total derivative d T by its "higher analogue" d 2 T , and the bundle ∧ 2 ÌE by the quotient bundle of ∧ 2 ÌE → E modulo its subbundle Î 2 E of 2-vertical vectors. These are the ideas we owe to W. M. Tulczyjew. What is more, the bundle ∧ 2 Ì ∧ 2 Ì * M can be viewed as double graded bundle in the sense of [22], so we get a picture completely analogous to (1.1) except for the fact that the morphisms are not isomorphisms. In this sense, our geometric approach to classical strings is based on morphisms of double graded bundles, although the canonical multisymplectic structure is behind. Let us note also that the graded bundles appear in the supergeometric context under the name graded manifolds or N-manifolds [38,40,48], in particular, in the Roytenberg's picture for Courant algebroids [38].
We want to stress that our framework is completely covariant and, what is more, not reduced to derivation of the Euler-Lagrange equations. We present the full picture, determining the phase space and the phase equations as the meeting point of the Lagrange and the Hamilton formalisms. The equations are generated geometrically from lagrangian submanifolds. In particular, on the Hamiltonian side we do not use, at least explicitly, any Poisson brackets.
Note finally that classical field theory is usually associated with the concept of a multisymplectic structure. The multisymplectic approach appeared first in the papers of the 'Polish school' [10,26,27,43]. Then, it was developed by Gotay, Isennberg, Marsden, and others in [13,14]. The original idea of the multisymplectic structure has been thoroughly investigated and developed by many contemporary authors, see e.g. [2,4,5,7,8]. The Tulczyjew triple in the context of multisymplectic field theories appeared recently in [3] and [31] (see also [46]). A similar picture, however with differences on the Hamiltonian side, one can find in [11] (see also [12,30]) and many others; it is not possible to list all achievements in this area.
As we use the canonical multisymplectic form on ∧ n Ì * M , the prototype of all the abstract definitions of such structure, our paper can be viewed also as an attempt to start again the discussion about what is the reasonable concept of an abstract multisymplectic structure (cf. [2,9,25,35]). We want to undertake this discussion in a separate paper.

Notation and local coordinates
Let M be a smooth manifold. We denote by τ M : ÌM → M the tangent vector bundle and by π M : Ì * M → M the cotangent vector bundle. Let τ : E → M be a vector bundle and π : E * → M the dual bundle. We use the following notations for tensor bundles: the modules of sections over C ∞ (M ): and the corresponding tensor algebras In particular, and By Î(E) we denote the subbundle of tangent vertical vectors: We introduce also the bundle Î 1 (E) of 1-vertical bivectors and its subbundle of 2-vertical bivectors Let (x µ ), µ = 1, . . . , n, be a coordinate system in M . We introduce the induced coordinate systems Formally, we should take only ν < µ, but we can use all pairs of indices, assuming p µλ = −p λµ andẋ νλ = −ẋ λµ . With this convention, we have the following representation of bivectors and bi-covectors: 1 2ẋ Let τ : E → M be a vector bundle and let (x µ , e a ) be an affine coordinate system on E, consisting of basic functions x µ (constant along fibers) and functions e a linear along fibers. We introduce the adapted coordinate systems In ÌÌM, we use the coordinates ( Consequently, the following representations of bivectors will be used: [36]). A standard definition is given by the graded commutator d where v i ∈ Ì u ÌN. By analogy, we define (cf. [1]) a graded differential operator of degree -1 as the graded commutator where i 2 T is a graded differential operator of degree -2 defined by the formula It is obvious that d T commutes (in the graded sense) with the exterior derivative: In local coordinates, for a 1-form ϕ = ϕ µ dx µ , we have i 2 T ϕ = 0 and It is easy to see that, for a differentiable mapping Ψ : M → N , and that a p-form α, p > 1, on N is closed if and only if and

Double vector bundles
A double vector bundle is a manifold with two compatible vector bundle structures. The compatibility conditions, in its original formulation (see [29,37]) are complicated, but in a recent work [21] one can find a very simple formulation of the compatibility condition. It is based on the observation that a vector bundle structure is encoded in its Euler vector field or, equivalently, in a multiplicative monoid R 0 = {r ∈ R : r 0} action on the total vector bundle manifold. In local coordinates, the Euler vector field X E for a vector bundle τ : E → M is given by A vector bundle morphism ϕ : E → F is characterized by the property ÌϕX E (e) = X F (ϕ(e)) for each e ∈ E. Two vector bundle structures are compatible if the corresponding Euler vector fields commute (equivalently, the monoid actions commute). A manifold with two compatible vector bundle structures is called a double vector bundle. A double vector bundle structure on a manifold K is represented by the commutative diagram of vector bundle morphisms: (4.1) For a double vector bundle, a zero section of one vector bundle structure is a subbundle of the second one. The projection of one vector bundle structure is a vector bundle morphism for the second one. In other words, the pairs (τ r ,τ r ) and (τ l ,τ l ) are vector bundle morphisms. The intersection of kernels of vector bundle projections τ r , τ l is a vector bundle over M , called the core of the double vector bundle. In other words, the core is a submanifold, where two Euler fields coincide. A mapping of double vector bundles is a double vector bundle morphism if it is a vector bundle morphism for both vector bundle structures. It follows that the image of the core of the domain is contained in the core of the co-domain. Note that the tangent and cotangent bundle of a vector bundle carry canonical double vector bundle structures.
The core is the bundle of co-vectors at the zero section, vanishing on vertical vectors (e a = 0, π a = 0). It can be identified with Ì * M .
A double vector bundle has its dual vector bundles with respect to the right and to the left vector bundle structures. It appears that right and left dual bundles are also double vector bundles. For a double vector bundle (4.1) with the core bundle C, the diagram for the right dual looks as follows (4.2) E * l is the core of K * r . For details see [29].

Bundle structures on bivectors
A vector space (vector bundle) structure is uniquely determined by the Euler vector field, however it can be also introduced by a pairing, i.e. by constructing the space of linear functions. We use this method to discuss vector bundle structures related to the fibration ∧ 2 ÌE over ∧ 2 ÌM. In the next section we shall discuss also the Euler vector field approach. It is easy to realize that the fibration ∧ 2 ÌE → ∧ 2 ÌM is not a canonical vector bundle structure. We will show that such structure exists in the quotient ∧ 2 ÌE Î 2 (E). In order to find a proper pairing, we begin with a co-vector valued pairing.
Let τ : E → M and ζ : F → M be vector bundles and let be a bilinear vector bundle mapping. δ can be considered a semi-basic 1-form on E × M F . In local coordinates, where (x µ , e a ), (x µ , f i ) are coordinates on E, F respectively and δ µai are functions on M . Using the formula (3.5) and coordinates as in (2.5), we obtain local expression for the function Proposition 5.1. The bundles are canonically isomorphic.

Proof.
Let be the canonical projections. We consider the mapping Since projections p E and p F are vector bundle morphisms, we have It is obvious that the kernel of this morphism is contained in Î 2 (E × M F ). It follows then from In the following, we denote by∧ 2 ÌE the quotient bundle ∧ 2 ÌE Î 2 (E). The above proposition implies thatd 2 T δ defines a pairing between∧ 2 ÌE and∧ 2 ÌF as bundles over ∧ 2 ÌM.
Such pairing (if not degenerate) can be used to define a vector bundle structure in the fibration We shall consider two special cases. where y νa are coordinates in ∧ 2 ÌE and y ν aµ are coordinates in ∧ 2 Ì(E * ⊗ M Ì * M ). We see from this formula and Proposition (5.1) that d 2 T δ projects to a function . Moreover, the set {g u : ∧ 2 Ìτ(u) = w} is a vector subspace in the vector space of all functions on (∧ 2 Ì(π ⊗ M π M )) −1 (w). We have proved the following.
Theorem 5.1. There is a canonical vector bundle structure on the fibration The coordinates (x µ , e a ,ẋ λν , y κb ,ė cd ) on ∧ 2 ÌE, introduced by the formula (2.5), induce coordinates (x µ , e a ,ẋ λν , y κb ) on∧ 2 ÌE. The canonical projection so that (e a , y κb ) are linear coordinates and the Euler vector field reads X∧2 TE = e a ∂ e a + y κb ∂ y κb .
Since it commutes with the standard Euler vector field X∧2 TE =ẋ λν ∂ẋλν + y κb ∂ y κb , the two vector bundle structures on∧ 2 ÌE are compatible, i.e. form a double vector bundle. The core of this double vector bundle is canonically isomorphic to ÌM ⊗ M E.

The second special case: E = ⊤M
In this case we replace the bundle E * ⊗ M Ì * M = Ì * M ⊗ M Ì * M by its subbundle ∧ 2 Ì * M .
We denote by ∆ 2 M the mapping δ TM restricted to ÌM × M ∧ 2 Ì * M : where i v α(w) = α(v, w). In local coordinates, Thus, the function d 2 T ∆ 2 M represents a 'pairing' between the fibrations which projects to an actual pairing of vector bundles

The mapping 'kappa' for bivectors
As it was stressed in the introduction, a fundamental role in analytical mechanics is played by the canonical isomorphism of double vector bundles which encodes the Lie algebroid structure of ÌM [17,24].
Let us consider a sequence of mappings Since ∧ 2 is bilinear and skew-symmetric, also Ì∧ 2 is bilinear with respect to the tangent vector bundle structures and skew-symmetric. It follows that the composition Ì ∧ 2 •(κ M × κ M ) is bilinear and skew-symmetric and consequently, defines a mapping which is a vector bundle morphism with respect to the vector bundle structures over ÌM (see [23]). It follows from the construction that The first is an immediate consequence of the fact that κ M intertwines projections τ TM and Ìτ M . In order to see that Î 2 (ÌM ) is contained in the kernel of κ 2 M , let us take two vertical vectors w 1 , w 2 ∈ ÎÌM with τ TM (w 1 ) = τ TM (w 2 ). The vectors κ M (w i ) have the properties: τ TM (κ M (w i )) = 0 and Ìτ M (κ M (w 1 )) = Ìτ M (κ M (w 2 )). It follows that in a local trivialization of ÌM, the vector w i can be represented by a curve in ÌM of the form t → (x(t), tv i ).
, which represents a zero vector.
In local coordinates, Now, we find a coordinate-free description of the kernel of κ 2 M as a vector bundle morphism over ÌM. As a tangent bundle of a vector bundle, Ì ∧ 2 ÌM is a double vector bundle ( [29]) and it follows that the zero section of the vector bundle Ìτ ∧2 M : Ì ∧ 2 ÌM → ÌM lies in the kernel of the projection τ ∧ 2 TM . Since κ 2 M respects the fibrations over ∧ 2 ÌM, we have ∧ 2 Ìτ M (u) = 0 ∈ ∧ 2 ÌM for u ∈ ker κ 2 M . As we have already noticed, Î 2 (ÌM ) ⊂ ker κ 2 M .
Thus, the kernel is completely characterized by its image in the vector bundle∧ 2 ÌÌM. It is a general fact for a double vector bundle 4.1 with the core bundle C that τ −1 r (0) is canonically isomorphic to E l × M C. It follows that

and the canonical projection τ ∧2
TM corresponds to the projection on the first component. Similarly, and we get the induced mapping It follows from the local formulae that and that the kernel of κ 2 M consists of bivectors in Î 1 (ÌM ), which represent symmetric elements in ÌM ⊗ M ÌM. It is now easy task to verify that κ 2 M induces the following commutative diagram of maps which projects onto a morphism of double vector bundles: (6.5)

Graded bundle approach
The question of the bundle structures in ∧ 2 ÌE can be approached differently following the concept of a graded bundle [22] which generalizes that of a vector bundle (see also [38,40]). The role of the Euler vector field is played by the weight vector field which in homogeneous coordinates is a linear diagonal vector field with weights being positive integers. An important observation is that graded bundles of degree 1 are just vector bundles, as weight vector fields are Euler vector field. There is also a nice interpretation of graded bundles in terms of smooth actions of the multiplicative monoid (R, ·) [21,22]. It is obvious, that τ 2 E : ∧ 2 ÌE → E is a vector bundle with the Euler vector field X ∧ 2 TE =ẋ µν ∂ ∂ẋ µν + y µa ∂ ∂y µa +ė bc ∂ ∂ė bc . (6.6) Another graded bundle structure is given by the weight vector field d 2 T X E which in coordinates reads d 2 T X E = e a ∂ ∂e a + y µd ∂ ∂y µd + 2ė bc ∂ ∂ė bc , (6.7) so that makes ∧ 2 ÌE into a graded bundle of degree 2. The coordinates (x µ ,ẋ µν ) are of degree 0, (e a , y µb ) are of degree 1, and (ė ab ) are of degree 2. The reduction to coordinates of degree ≤ 1 gives the vector bundle∧ 2 Ìτ :∧ 2 ÌE → ∧ 2 ÌM. More precisely, the canonical inclusions of function algebras where A k is the algebra of polynomials in homogeneous functions of degree ≤ k, induce the sequence of bundle projections where the first is an affine, and the second is a vector bundle projection. Elements od A 1 were interpreted in section 5.1 as sections of certain quotient bundle of the bundle∧ 2 ÌF → ∧ 2 ÌM. The first projection coincides with the canonical projection from ∧ 2 ÌE to the quotient spacē Note that the vector field d 2 T X E is the "bi-tangent" lift of the vector field X E to the bundle of bivectors. Recall that the tangent (complete) lift of a vector field Y on a manifold M to the vector field d T Y on ÌM can be defined as In the context of bivectors we can do the same using ∧ 2 Ì functor and κ 2 Applying the above formula to the Euler vector field X E = e a ∂ ∂e a of the bundle τ we get precisely (6.7). It is easy to check that the vector fields X ∧ 2 TE and d 2 T X commute, therefore The weight vector fields X ∧ 2 TE and d 2 T X define a bi-gradation in which the coordinates (x µ , e a ,ẋ µν , y µa ,ė ab ) are of the bi-degrees (0, 0), (0, 1), (1, 0),(1,1), and (1, 2), respectively.
Moreover, since X ∧ 2 TE is projectable on∧ 2 ÌE, its projection together with the projection of d 2 T X E define the double vector bundle structure on∧ 2 ÌE: We just forget the coordinates of degree (1,2).
Quite similarly, if we start with a graded bundle G of degree 2 with coordinates (x µ , y a , z w ) of degrees, respectively, 0, 1, 2 with respect to the weight vector field X G = y a ∂ y a + 2z w ∂ z w , then ÌG is a double graded bundle with respect to the Euler vector field X TG =ẋ µ ∂ẋµ +ẏ a ∂ẏa +ż w ∂żw and the tangent lift d T X G = y a ∂ y a + 2z w ∂ z w +ẏ a ∂ẏa + 2ż w ∂żw .

The Lagrangian side of the triple
The introduced in Section 5.2 function is a degenerate pairing between fibrations We have also the canonical pairing With this pairings we define a relation which is the dual relation to the converse (called sometimes also: inverse or transpose) of κ 2 M : . In local coordinates, with the following representations of p, w, u: we have , in view of (6.3), reads It follows that the relation α 2 M is given by p ρ = −y η ηρ , f λκ = −p λκ (7.3) and that it can be considered a mapping which projects to a mappingᾱ which is a morphism of double vector bundles (but not an isomorphism). In other words, where (x µ , p λκ ,ẋ νσ , y η θρ ,ṗ γδǫζ ) are coordinates on (9.5) The above diagram consists of double graded bundle morphisms which simultaneously preserve the presymplectic structures. Its reduction to degree 1 gives a diagram of morphisms of double vector bundles 10 Phase dynamics

Lagrangian dynamics and Euler-Lagrange equations
The way of obtaining the implicit phase dynamics D, as a submanifold of ∧ 2 Ì ∧ 2 Ì * M , from a Lagrangian L : ∧ 2 ÌM → R is now standard: we take the lagrangian submanifold L(L) = (dL)(∧ 2 ÌM) of the cotangent bundle Ì * ∧ 2 ÌM and then its inverse image D = which is said to be generated by the Morse family H.

An example
In the dynamics of strings, the manifold of infinitesimal configurations is ∧ 2 ÌM, where M is the space time with the Lorentz metric g. This metric induces a scalar product h in fibers of where h µνκλ = g µκ g νλ − g µλ g νκ .
The Lagrangian is a function of the volume with respect to this metric, the so called Nambu-Goto Lagrangian The Euler-Lagrange equations (10.1)-(10.2), applied for surfaces being graphs (x, y) → (x, y, z(x, y)), in this case reaḋ