Table Of ContentCovariant Hamiltonian formalism for the calculus of
variations with several variables: Lepage–Dedecker
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versus de Donder–Weyl
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n Fr´ed´eric HE´LEIN∗
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J Institut de Mathmatiques de Jussieu, UMR 7586,
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Universit Denis Diderot–Paris 7, Site de Chevaleret,
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16 rue Clisson 75013 Paris (France)
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v
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4 Joseph KOUNEIHER†
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LUTH, CNRS UMR 8102
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Observatoire de Paris - section Meudon
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0 5 Place Jules Janssen
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92195 Meudon Cedex
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- Universit´e Paris 7
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t
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m February 7, 2008
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a Abstract
The main purpose in the present paper is to build a Hamiltonian theory for fields which is
consistent withtheprinciplesofrelativity. Forthisweconsider detailedgeometricpictures
of Lepage theories in the spirit of Dedecker and try to stress out the interplay between
the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one.
One of the main points is the fact that the Legendre transform in the dDW approach is
replaced by a Legendre correspondence in the LP theory1.
∗[email protected]
†[email protected]
1This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is de-
generate.
1 Introduction
1.1 Presentation
Multisymplectic formalisms are finite dimensional descriptions of variational problems
with several variables (or field theories for physicists) analogue to the well-known Hamil-
tonian theory of point mechanics. For example consider on the set of maps u : Rn −→ R
a Lagrangian action of the type
L[u] = L(x,u(x),∇u(x))dx1···dxn.
ZRn
Then it is well-known that the maps which are critical points of L are characterized by
the Euler–Lagrange equation ∂ ∂L = ∂L. By analogy with the Hamiltonian theory
∂xµ (cid:16)∂(∂µu)(cid:17) ∂u
we can do the change of variables pµ := ∂L and define the Hamiltonian function
∂(∂µu)
∂u
H(x,u,p) := pµ −L(x,u,∇u),
∂xµ
wherehere∇u = ∂u isafunctionof(x,u,p)definedimplicitlybypµ := ∂L (x,u,∇u).
∂xµ ∂(∂µu)
Then the Euler-L(cid:0)agra(cid:1)nge equation is equivalent to the generalized Hamilton system of
equations
∂u ∂H
= (x,u,p)
∂xµ ∂pµ
∂pµ ∂H (1)
= − (x,u,p).
∂xµ ∂u
Xµ
This simple observation is the basis of a theory discovered by T. de Donder [3] and H.
Weyl [18] independently in 1935. This theory can be formulated in a geometric setting, an
analogue of the symplectic geometry, which is governed by the Poincar´e–Cartan n-form
θ := eω + pµdu∧ω (where ω := dx1 ∧···∧ dxn and ω := ∂ ω) and its differential
µ µ µ
Ω := dθ, often called multisymplectic (or polysymplectic form).
Although similar to mechanics this theory shows up deep differences. In particular there
exist other theories which are analogues of Hamilton’s one as for instance the first histor-
ical one, constructed by C. Carath´eodory in 1929 [2]. In fact, as realized by T. Lepage in
1936 [14], there are infinitely many theories, due to the fact that one could fix arbitrary
the value of some tensor in the Legendre transform (see also [16], [5]). Much later on,
in 1953, P. Dedecker [4] built a geometrical framework in which all Lepage theories are
embedded. The present paper, which is a continuation of [7], is devoted to the study
of the Lepage–Dedecker theory. We also want to compare this formalism with the more
popular de Donder–Weyl theory.
First recall that the range of application of the de Donder–Weyl theory is restricted in
principle to variational problems on sections of a bundle F. The right framework for it, as
expounded e.g.in [6], consists in using the affine first jet bundle J1F and its dual (J1)∗F
as analogues of the tangent and the cotangent bundles for mechanics respectively. For
non degenerate variational problems the Legendre transform induces a diffeomorphism
between J1F and (J1)∗F. In contrast the Lepage theories can be applied to more general
situations but involve, in general, many more variables and so are more complicated to
deal with, as noticed in [13]. This is probably the reason why most papers on the subject
focus on the de Donder–Weyl theory, e.g.[12], [6]. The general idea of Dedecker in [4]
for describing Lepage’s theories is the following: if we view variational problems as being
defined on n-dimensional submanifolds embedded in a (n+k)-dimensional manifold N,
then what plays the role of the (projective) tangent bundle to space-time in mechanics is
the Grassmannbundle GrnN of oriented n-dimensional subspaces of tangent spaces to N.
The analogueof thecotangent bundle inmechanics isΛnT∗N. Notethat dimGrnN = n+
k+nk so that dimΛnT∗N = n+k+(n+k)! isstrictly larger thandimGrnN+1unless n = 1
n!k!
(classical mechanics) or k = 1 (submanifolds are hypersurfaces). This difference between
thedimensionsreflectsthemultiplicityofLepagetheories: asshownin[4],wesubstituteto
the Legendre transform a Legendre correspondence which associates to each n-subspace
T ∈ GrnN (a “generalized velocity”) an affine subspace of ΛnT∗N called pseudofibre
q q
by Dedecker. Then two points in the same pseudofiber do actually represent the same
physical (infinitesimal) state, so that the coordinates on ΛnT∗N, called momento¨ıdes by
Dedecker do not represent physically observable quantities. In this picture any choice of
a Lepage theory corresponds to a selection of a submanifold of ΛnT∗N, which — when
the induced Legendre transform is invertible — intersects transversally each pseudofiber
at one point (see Figure 1.1): so the Legendre correspondence specializes to a Legendre
transform. For instance the de Donder–Weyl theory can be recovered in this setting by
the restriction to some submanifold of ΛnT∗N (see Section 2.2).
pseudofibres
a choice of a
Lepage theory
Figure 1: Pseudofibers which intersect a submanifold corresponding to the choice of a Lepage theory
In [7] and in the present paper we consider a geometric pictures of Lepage theories in the
spirit of Dedecker and we try to stress out the interplay between the Lepage–Dedecker
description and the de Donder–Weyl one. Roughly speaking a comparison between these
two points of view shows up some analogy with some aspects of the projective geometry,
for which there is no perfect system of coordinates, but basically two: the homogeneous
ones, more symmetric but redundant (analogue to the Dedecker description) and the local
ones (analogue to the choice of a particular Lepage theory like e.g.the de Donder–Weyl
one). Note that both points of view are based on the same geometrical framework, a
multisymplectic manifold:
Definition 1.1 Let M be a differential manifold. Let n ∈ N be some positive integer. A
smooth (n+1)-form Ω on M is a multisymplectic form if and only if
(i) Ω is non degenerate, i.e.∀m ∈ M, ∀ξ ∈ T M, if ξ Ω = 0, then ξ = 0
m m
(ii) Ω is closed, i.e.dΩ = 0.
AnymanifoldMequipped with amultisymplectic form Ωwill becalled amultisymplectic
manifold.
For the de Donder–Weyl theory M is (J1)∗F and for the Lepage–Dedecker theory M
is ΛnT∗N. In both descriptions solutions of the variational problem correspond to n-
dimensional submanifolds Γ (analogues of Hamiltonian trajectories: we call them Hamil-
tonian n-curves) and are characterized by the Hamilton equation X Ω = (−1)ndH,
where X is a n-multivector tangent to Γ, H is a (Hamiltonian) function defined on M
and by “ ” we mean the interior product.
In Section 2 we present a complete derivation of the (Dedecker) Legendre correspondence
and of the generalized Hamilton equations. We use a method that does not rely on any
trivialization or connection on the Grassmannian bundle. A remarkable property, which
is illustrated in this paper through the examples given in Paragraph 2.2.2, is that when
n and k are greater than 2, the Legendre correspondence is generically never degenerate.
The more spectacular example is when the Lagrangian density is a constant function —
themost degenerate situationone canthinkabout— thentheLegendre correspondence is
well-defined almost everywhere except precisely along the de Donder–Weyl submanifold.
We believe that such a phenomenon was not noticed before; it however may be useful
when one deals for example with the bosonic string theory with a skewsymmetric 2-form
on the target manifold (a “B-field”, as discussed in [7] and in subsection 2.2, example 5)
or with the Yang–Mills action in 4 dimensions with a topological term in the Lagrangian:
then the de Donder–Weyl formalism may fail but one can cure this degenerateness by
using another Lepage theory or by working in the full Dedecker setting.
In this paper we also stress out another aspect of the (Dedecker) Legendre correspon-
dence: one expects that the resulting Hamiltonian function on ΛnT∗N should satisfy
some condition expressing the “projective” invariance along each pseudofiber. This is
indeed the case. On the one hand we observe in Section 2.1 that any smoothly contin-
uous deformation of a Hamiltonian n-curve along directions tangent to the pseudofibers
remains a Hamiltonian n-curve2 (Corollary 2.1). On the other hand we give in Section
4.3 an intrinsic characterization of the subspaces tangent to pseudofibers. This motivates
2A property quite similar to a gauge theory behavior although of different meaning. Here we are
interested by desingularizing the theory and avoid the problems related to the presence of a constraints.
the definition given in Section 3.3 of the generalized pseudofiber directions on any multi-
symplectic manifold.
Beside these properties in this paper and in its companion paper [9] we wish to address
other kind of questions related to the physical gain of these theories: the main advantage
of multisymplectic formalisms is to offer us a Hamiltonian theory which is consistent with
the principles of Relativity, i.e.being covariant. Recall for instance that for all the mul-
tisymplectic formalisms which have been proposed one does not need to use a privilege
time coordinate. One of our ambitions in this paper was to try to extend this democracy
between space and time coordinates to the coordinates on fiber manifolds (i.e.along the
fields themselves). This is quite in the spirit of the Kaluza–Klein theory and its modern
avatars: 11-dimensional supergravity, string theory and M-theory. This concern leads us
naturallytoreplace deDonder–Weylby theDedecker theory. Inparticular wedo notneed
in our formalism to split the variables into the horizontal (i.e.corresponding to space-time
coordinates) and vertical (i.e.non horizontal) categories.
Moreover we may think that we start from a (hypothetical) geometrical model where
space-time and fields variables would not be distinguished a priori and then ask how to
make sense of a space-time coordinate function (that we call a “r-regular” in Section 3.2)
? A variant of this question would be how to define a constant time hypersurface (that
we call a “slice” in Section 3.2) without referring to a given space-time background ?
We propose in Section 3.2 a definition of r-regular functions and of slices which, roughly
speaking, requires a slice to be transversal to all Hamiltonian n-curves. Here the idea
is that the dynamics only (i.e.the Hamiltonian equation) should determine what are the
slices. We give in Section 4.2 a characterization of these slices in the case where the
multisymplectic manifold is ΛnT∗N.
These questions are connected to the concept of observable functionals over the set of
solutions of the Hamilton equation. First because by using a codimension r slice Σ and
an (n−r)-form F on the multisymplectic manifold one can define such a functional by in-
tegrating F over the the intersection of Σ with a Hamiltonian curve. And second because
one is then led to impose conditions on F in such a way that the resulting functional
carries only dynamical information. The analysis of these conditions is the subject of
our companion paper [9]. And we believe that the conditions required on these forms
are connected with the definitions of r-regular functions given in this paper, although we
have not completely elucidated this point.
Lastly in a future paper [10] we investigate gauge theories, addressing the question of how
to formulate a fully covariant multisymplectic for them. Note that the Lepage–Dedecker
theory expounded here does not answer this question completely, because a connection
cannot be seen as a submanifold. We will show there that it is possible to adapt this
theory and that a convenient covariant framework consists in looking at gauge fields
as equivariant submanifolds over the principal bundle of the theory, i.e.satisfying some
suitable zeroth and first order differential constraints.
1.2 Notations
The Kronecker symbol δµ is equal to 1 if µ = ν and equal to 0 otherwise. We shall also
ν
set
δµ1 ... δµ1
ν1 νp
δνµ11······νµpp := (cid:12)(cid:12) ... ... (cid:12)(cid:12).
(cid:12)(cid:12)(cid:12) δνµ1p ... δνµpp (cid:12)(cid:12)(cid:12)
In most examples, ηµν is a constant me(cid:12)(cid:12)tric tensor on R(cid:12)(cid:12) n (which may be Euclidean or
Minkowskian). The metric on his dual space his ηµν. Also, ω will often denote a volume
form on some space-time: in local coordinates ω = dx1∧···∧dxn and we will use several
times the notation ω := ∂ ω, ω := ∂ ∧ ∂ ω, etc. Partial derivatives ∂ and
µ ∂xµ µν ∂xµ ∂xν ∂xµ
∂ will be sometime abbreviated by ∂ and ∂α1···αn respectively.
∂pα1···αn µ
When an index or a symbol is omitted in the middle of a sequence of indices or symbols,
[
we denote this omission by . For example ai1···ip···in := ai1···ip−1ip+1···in, dxα1 ∧···∧dxαµ ∧
···∧dxαn := dxα1 ∧···∧dxαµ−1 ∧dxαµ+1 ∧···∧dxαn.
b
b
If N is a manifold and FN a fiber bundle over N, we denote by Γ(N,FN) the set of
smooth sections of FN. Lastly we use the following notations concerning the exterior
algebra of multivectors and differential forms. If N is a differential N-dimensional man-
ifold and 0 ≤ k ≤ N, ΛkTN is the bundle over N of k-multivectors (k-vectors in short)
and ΛkT⋆N is the bundle of differential forms of degree k (k-forms in short). Setting
ΛTN := ⊕N ΛkTN and ΛT⋆N := ⊕N ΛkT⋆N, there exists a unique duality evaluation
k=0 k=0
map between ΛTN and ΛT⋆N such that for every decomposable k-vector field X, i.e.of
the form X = X ∧···∧X , and for every l-form µ, then hX,µi = µ(X ,···,X ) if k = l
1 k 1 k
and = 0 otherwise. Then interior products and are operations defined as follows.
If k ≤ l, the product : Γ(N,ΛkTN)×Γ(N,ΛlT⋆N) −→ Γ(N,Λl−kT⋆N) is given by
hY,X µi = hX ∧Y,µi, ∀(l−k)-vector Y.
And if k ≥ l, the product : Γ(N,ΛkTN)×Γ(N,ΛlT⋆N) −→ Γ(N,Λk−lTN) is given
by
hX µ,νi = hX,µ∧νi, ∀(k −l)-form ν.
2 The Lepage–Dedecker theory
We expound here a Hamiltonian formulation of a large class of second order variational
problems in an intrinsic way. Details and computations in coordinates can be found in
[12], [7].
2.1 Hamiltonian formulation of variational problems with sev-
eral variables
2.1.1 Lagrangian formulation
The category of Lagrangian variational problems we start with is described as follows.
We consider n,k ∈ N∗ and a smooth manifold N of dimension n+k; N will be equipped
with a closed nowhere vanishing “space-time volume” n-form ω. We define
• the Grassmannian bundle GrnN, it is the fiber bundle over N whose fiber over
q ∈ N is GrnN, the set of all oriented n-dimensional vector subspaces of T N.
q q
• the subbundle GrωN := {(q,T) ∈ GrnN/ω > 0}.
q|T
• the set Gω, it is the set of all oriented n-dimensional submanifolds G ⊂ N, such
that ∀q ∈ G, T G ∈ GrωN (i.e.the restriction of ω on G is positive everywhere).
q q
Lastly we consider any Lagrangian density L, i.e.a smooth function L : GrωN 7−→ R.
Then the Lagrangian of any G ∈ Gω is the integral
L[G] := L(q,T G)ω (2)
q
Z
G
We say that a submanifold G ∈ Gω is a critical point of L if and only if, for any compact
K ⊂ N, G∩K is a critical point of L [G] := L(q,T G)ω with respect to variations
K G∩K q
with support in K. R
It will be useful to represent GrnN differently, by means of n-vectors. For any q ∈ N, we
define DnN to be the set of decomposable n-vectors3, i.e.elements z ∈ ΛnT N such that
q q
there exists n vectors z ,...,z ∈ T N satisfying z = z ∧···∧z . Then DnN is the fiber
1 n q 1 n
bundle whose fiber at each q ∈ N is DnN. Moreover the map
q
DnN −→ GrnN
q q
z ∧···∧z 7−→ T(z ,···,z ),
1 n 1 n
where T(z ,···,z ) is the vector space spanned and oriented by (z ,···,z ), induces a
1 n 1 n
diffeomorphism between DnN \{0} /R∗ and GrnN. If we set also DωN := {(q,z) ∈
q + q q
DnN/ω (z) = 1}, the sam(cid:0)e map allow(cid:1) us also to identify GrωN with DωN.
q q q q
This framework includes a large variety of situations as illustrated below.
Example 1 — Classical point mechanics — The motion of a point moving in a manifold
Y can be represented by its graph G ⊂ N := R × Y. If π : N −→ R is the canonical
projection and t is the time coordinate on R, then ω := π∗dt.
3another notationfor this set would be DΛnT N, for it reminds that it is a subset of ΛnT N, but we
q q
have chosen to lighten the notation.
Example 2 — Maps between manifolds — We consider maps u : X −→ Y, where X
and Y are manifolds of dimension n and k respectively and X is equipped with some non
vanishingvolumeformω. AfirstorderLagrangiandensitycanrepresentedasafunctionl :
TY⊗ T⋆X 7−→ R, where TY⊗ T⋆X := {(x,y,v)/(x,y) ∈ X×Y,v ∈ T Y⊗T∗X}.
X×Y X×Y y x
(We use here a notation which exploits the canonical identification of T Y ⊗ T∗X with
y x
the set of linear mappings from T X to T Y). The action of a map u is
x y
ℓ[u] := l(x,u(x),du(x))ω.
Z
X
In local coordinates xµ such that ω = dx1∧···∧dxn, critical points of ℓ satisfy the Euler-
Lagrange equation n ∂ ∂l (x,u(x),du(x)) = ∂l (x,u(x),du(x)), ∀i = 1,···,k.
µ=1 ∂xµ ∂vi ∂yi
(cid:16) µ (cid:17)
Then we set N :=PX × Y and denoting by π : N −→ X the canonical projection,
we use the volume form ω ≃ π∗ω. Any map u can be represented by its graph G :=
u
{(x,u(x))/x ∈ X} ∈ Gω, (and conversely if G ∈ Gω then the condition ω > 0 forces G
|G
to be the graph of some map). For all (x,y) ∈ N we also have a diffeomorphism
T Y ⊗T∗X −→ Grω N ≃ Dω N
y x (x,y) (x,y)
v 7−→ T(v),
whereT(v)isthegraphofthelinearmapv : T X −→ T Y. ThenifwesetL(x,y,T(v)) :=
x y
l(x,y,v), the action defined by (2) coincides with ℓ.
Example 3 — Sections of a fiber bundle — This is a particular case of our setting, where
N is the total space of a fiber bundle with base manifold X. The set Gω is then just the
set of smooth sections.
2.1.2 The Legendre correspondence
Now we consider the manifold ΛnT∗N and the projection mapping Π : ΛnT∗N −→ N.
We shall denote by p an n-form in the fiber ΛnT∗N. There is a canonical n-form θ called
q
thePoincar´e–Cartanformdefined onΛnT∗N asfollows: ∀(q,p) ∈ ΛnT∗N, ∀X ,···,X ∈
1 n
T (ΛnT∗N),
(q,p)
θ (X ,···,X ) := p(Π X ,···,Π X ) = hΠ X ∧···∧Π n,pi,
(q,p) 1 n ∗ 1 ∗ n ∗ 1 ∗
where Π X := dΠ (X ). If we use local coordinates (qα) on N, then a basis
∗ µ (q,p) µ 1≤α≤n+k
of ΛnT∗N is the family (dqα1 ∧···∧dqαn) and we denote by p the
q 1≤α1<···<αn≤n+k α1···αn
coordinates on ΛnT∗N in this basis. Then θ writes
q
θ := p dqα1 ∧···∧dqαn. (3)
α1···αn
1≤α1<·X··<αn≤n+k
Its differential is the multisymplectic form Ω := dθ and will play the role of generalized
symplectic form.
In order to build the analogue of the Legendre transform we consider the fiber bundle
GrωN × ΛnT∗N := {(q,z,p)/q ∈ N,z ∈ GrωN ≃ DωN,p ∈ ΛnT∗N} and we denote
N q q q
by Π : GrωN × ΛnT∗N −→ N the canonical projection. To summarize:
N
b GrωN × ΛnT∗NΠH //ΛnT∗N oo ı M
N(cid:15)(cid:15)ΠL RRRRRRRΠbRRRRRRRR)) (cid:15)(cid:15)ΠzzuuuuuuΠu|uMuu
GrnN oo GrωN //N
ı
We define on GrωN × ΛnT∗N the function
N
W(q,z,p) := hz,pi−L(q,z).
Note that for each (q,z,p) there a vertical subspace V ⊂ T (GrωN × ΛnT∗N),
(q,z,p) (q,z,p) N
which is canonically defined as the kernel of
dΠ : T (GrωN × ΛnT∗N) −→ T N.
(q,z,p) (q,z,p) N q
We can further split Vb ≃ T DωN ⊕ T ΛnT∗N, where T DωN ≃ KerdΠH and
(q,z,p) z q p q z q (q,z,p)
T ΛnT∗N ≃ KerdΠL . Then, for any function F defined on GrωN × ΛnT∗N, we de-
p q (q,z,p) N
note respectively by ∂F/∂z(q,z,p) and ∂F/∂p(q,z,p) the restrictions of the differential4
dF on respectively T DωN and T ΛnT∗N.
(q,z,p) z q p q
Instead of a Legendre transform we shall rather use a Legendre correspondence: we write
∂W
(q,z) ←→ (q,p) if and only if (q,z,p) = 0. (4)
∂z
Let us try to picture geometrically the situation (see figure 2.1.2): DωN is a smooth
q
ω
DqωN TzDqN
z
n
Λ T N
q
Figure 2: T DωN is a vector subspace of ΛnT N
z q q
submanifold of dimension nk of the vector space ΛnT N, which is of dimension (n+k)!;
q n!k!
4Howeverinordertomakesenseof“∂F/∂q(q,z,p)”wewouldneedtodefinea“horizontal”subspaceof
T (GrωN × ΛnT∗N),whichrequiresforinstancetheuseofaconnectiononthebundle GrωN×
(q,z,p) N N
ΛnT∗N −→N. Indeedsucha horizontalsubspaceprescribesa inertiallaw onN, suchalaw wouldhave
a sense on a Galilee or Minkowski space-time but not in general relativity.
T DωN isthusavectorsubspace ofΛnT N. And ∂L(q,z)or ∂W(q,z,p)canbeunderstood
z q q ∂z ∂z
aslinearformsonT DωN whereasp ∈ ΛnT∗N asalinearformonΛnT N. Sothemeaning
z q q q
oftherighthandside of(4) isthattherestrictionofpatT DωN coincides with ∂L(q,z,p):
z q ∂z
∂L
p = (q,z). (5)
|TzDqωN ∂z
Given (q,z) ∈ GrωN we define the enlarged pseudofiber in q to be:
∂W
P (z) := {p ∈ ΛnT∗N/ (q,z,p) = 0}.
q q ∂z
In other words, p ∈ P (z) if it is a solution of (5). Obviously P (z) is not empty; moreover
q q
given some p ∈ P (z),
0 q
p ∈ P (z), ⇐⇒ p −p ∈ T DωN ⊥ := {p ∈ ΛnT∗N/∀ζ ∈ T DωN,p(ζ) = 0}. (6)
1 q 1 0 z q q z q
(cid:0) (cid:1)
So P (z) is an affine subspace of ΛnT∗N of dimension (n+k)!−nk. Note that in case where
q q n!k!
n = 1 (the classical mechanics of point) then dimP (z) = 1: this is due to the fact that we
q
arestillfreetofixarbitrarilythemomentum componentdualtothetime(i.e.theenergy)5.
We now define
P := P (z) ⊂ ΛnT∗N, ∀q ∈ N
q q q
z∈[DωN
q
and we denote by P := ∪ P the associated bundle over N. We also let, for all
q∈N q
(q,p) ∈ ΛnT∗N,
Z (p) := {z ∈ GrωN/p ∈ P (z)}.
q q q
It is clear that Z (p) 6= ∅ ⇐⇒ p ∈ P . Now in order to go further we need to choose some
q q
submanifold M ⊂ P , its dimension is not fixed a priori.
q q
Legendre Correspondence Hypothesis — We assume that there exists a subbundle
manifold M ⊂ P ⊂ ΛnT∗N over N where dimM =: M such that,
• for all q ∈ N the fiber M is a smooth submanifold, possibly with boundary, of
q
dimension 1 ≤ M −n−k ≤ (n+k)!
n!k!
• for any (q,p) ∈ M, Z (p) is a non empty smooth connected submanifold of GrωN
q q
• if z ∈ Z (p), then we have Z (p) = {z ∈ DωN/∀p˙ ∈ T M ,hz −z ,p˙i = 0}.
0 q q q p q 0
5a simple but more interesting example is provided by variational problems on maps u :
R2 −→ R2. Then one is led to the multisymplectic manifold Λ2T⋆R4. And given any
(q,z) ∈ GrωR4 the enlarged pseudofiber P (z) ⊂ Λ2T⋆R4 is an affine plane parallel to
q
R v1v2−v2v1 dx1∧dx2−ǫ vjdyi∧dxν +dy1∧dy2 ⊕ Rdx1 ∧ dx2, where (using the notations of
1 2 1 2 ij ν
Ex(cid:2)a(cid:0)mple 2) T(v(cid:1))=z. For details see Paragraph2.2.2. (cid:3)
