arXiv:0704.0433v3  [math-ph]  13 Oct 2008
A variational formulation of electrodynamics with external sources
Antonio De Nicola
CMUC, Department of Mathematics,
University of Coimbra,
3001-454 Coimbra, Portugal
antondenicola@gmail.com
W lodzimierz M. Tulczyjew
Associated with
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Italy
Valle San Benedetto, 2 – 62030 Montecavallo (MC), Italy
tulczy@libero.it
Abstract
We present a variational formulation of electrodynamics using de Rham even and odd differential
forms. Our formulation relies on a variational principle more complete than the Hamilton principle
and thus leads to field equations with external sources and permits the derivation of the constitutive
relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment
of different types of boundary problems in an unified way. In particular we obtain a smooth transition
to the infinitesimal version by using a current with a one point support.
2000 Mathematics Subject Classification: 49S05, 70S05.
Keywords: variational principles, electrodynamics.
Introduction
A general framework for variational formulations of physical theories was presented in [1]. Appli-
cations to statics and dynamics of mechanical systems appear in [2, 3]. In this paper we present a
variational formulation of electrodynamics based on that framework. Our work is related to the gen-
eral formulation of linear field theories in a symplectic framework contained in [4] and to the earlier
formulations of electrodynamics contained in [5, 6]. Some of the results contained in this paper were
announced in [7].
We are presenting a variational formulation of electrodynamics in an intrinsic, frame independent
fashion in the affine Minkowski space-time using de Rham odd and even differential forms ([8, 9, 10])
which permit the rigorous formulations of electrodynamics and the description of the transformation
properties of electromagnetic fields relative to reflections (see [5, 6]). Observable quantities like the
charge contained in a compact volume or the flux of the electromagnetic field through a surface require
the integration of differential forms. The use of odd quantities is not just a matter of elegance. Even
if the space-time is assumed to be orientable, the alternative approach of using standard differential
forms and the Hodge star operator require the choice of a specific orientation, i. e. the addition of an
extrinsic structure. Other recent treatments of classical electrodynamics using odd and even differential
forms can be found in [11, 12, 13].
Our construction is an example of application of the general variational framework [1]. Similar
constructions are needed for the variational formulation of a general field theory. The linearity of
classical electrodynamics and the choice of formulating it on the affine Minkowski space makes our
presentation simpler. Relying on a variational principle more complete than the Hamilton principle
our formulation leads to field equations with external sources. This variational principle also permits
http://arxiv.org/abs/0704.0433v3
the derivation of the constitutive relations which are usually postulated separately since the variations
normally considered are not general enough to derive them from the variational principle.
We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of
different types of boundary problems in an unified way. As an example we obtain a smooth transition to
the infinitesimal version by using a current with a one point support. De Rham currents are essentially
objects dual to differential forms.
The present paper is organized in the following way. In the first part we provide the geometric
structures needed for the rigorous formulation of electrodynamics. Part of this material is based on [5]
and is briefly reported here for the sake of completeness. We recall the Cartan calculus for odd and
even differential forms and their integration theory over odd and even de Rham currents.
The second part contains the main results. We start with the construction of a suitable space of
fields for electrodynamics (not a differential manifold) and a construction of tangent and cotangent
vectors. A convenient representation of these objects is introduced. The definition of the space of
fields is inspired by a similar construction suited for the statics of continuous media which is contained
in the final section of [1]. In Section 3 we formulate a variational principle for electrodynamics similar
to the virtual action principle of analytical mechanics with external forces and boundary terms and
derive the field equations which include the constitutive relations in addition to Maxwell’s equations.
The boundary problem in a finite domain is treated in Section 4. Section 5 contains the Lagrangian
formulation of electrodynamics. The Legendre transformation and the Hamiltonian formulation of
electrodynamics in Section 6 conclude the paper.
A. Preliminaries
Here we provide the geometric structures needed for the rigorous formulation of electrodynamics in
Part B. The material in Sections 1, 2, and 4 is based on [5] and is briefly recalled here for the sake of
completeness. Nevertheless, we add a more explicit presentation of some useful details.
1.Orientations of vector spaces and vector subspaces.
Let V be a vector space of dimension m 6= 0. We denote by F(V ) the space of linear iso-
morphisms from V to Rm called frames. It is known that F(V ) is a homogeneous space with re-
spect to the natural group action of the general linear group GL(m,R) in F(V ). Let GL+(m,R)
and GL−(m,R) be the two connected components of the group GL(m,R). The set of orientations
O(V ) = F(V )
GL+(m,R) has two elements. This set is a homogeneous space for the quotient group
H(m,R) = GL(m,R)
GL+(m,R). The sets E = GL+(m,R) and P = GL−(m,R) are the elements of
the quotient group H(m,R) which is the group of permutations of the two elements of O(V ). There is
an ordered base (e1, e2, . . . , em) of V associated with each frame ξ in a obvious way.
Let W ⊂ V be a subspace of a vector space V . The subspace has the set O(W ) of orientations called
inner orientations of W . Orientations of the quotient space V
W are called outer orientations of W .
An outer orientation o′′ of W can be determined by specifying an inner orientation o of W together
with an orientation o′ of V . Let (e1, . . . , en) be the base of W associated with a frame ξ ∈ o. This base
can be completed to a base (e′1, . . . , e
m) of V with (e
1, . . . , e
n) = (e1, . . . , en). The extended base can
be chosen to be associated with a frame ξ′ ∈ o′. Let π:V → V
W be the canonical projection. The
sequence
(e′′1 , . . . , e
m−n) = (π(e
n+1), . . . , π(e
m)) (1)
is a base of V
W . It determines an orientation o′′ of V
W . Hence an outer orientation of W . The
outer orientation o′′ of W constructed from o ∈ O(W ) and o′ ∈ O(V ) is the same as the orientation
constructed from Po and Po′.
We have introduced inner orientation of subspaces of dimension different from zero and outer orien-
tation of subspaces of codimension different from zero. Integration theory of differential forms requires
the possibility of assigning inner orientations to the subspace W = {0} ⊂ V and outer orientations to
the subspace W = V . Two possible orientations are assigned to the subspace W = {0} ⊂ V one of
which is distinguished. The distinguished orientation is denoted by (+) and the other orientation is
denoted by (−). In agreement with the conventions established for orientation the outer orientations
of the subspace W = {0} ⊂ V are the orientations of V . An outer orientation of W = {0} can be
specified in terms of an inner orientation and an orientation of V . If the inner orientation is (+) and the
orientation of V is o, then o is the outer orientation of W . The orientation Po is the outer orientation
derived from (−) and o. The subspace W = V has a distinguished outer orientation defined as an
orientation of V
2.Multicovectors and multivectors.
A q - covector in a vector space V is a mapping a :×qV ×O(V ) → R. This mapping is q - linear and
totally antisymmetric in its vector arguments. A q - covector a is said to be even, if
a(v1, v2, . . . , vq, Po) = a(v1, v2, . . . , vq, o). (2)
It is said to be odd, if
a(v1, v2, . . . , vq, Po) = −a(v1, v2, . . . , vq, o). (3)
The vector space of even q - covectors will be denoted by ∧qeV
∗ and the space of odd q - covectors will
be denoted by ∧qoV
∗. We will use the symbol ∧qpV
∗ to denote either of the spaces in constructions valid
for both parities. The index p with the two possible values e and o will be used on other occasions.
For the definition of exterior product of (even and odd) multicovectors we refer to [5]. Here we just
recall that if two multicovectors a and a′ are even or both are odd, the product a∧a′ is even. In other
cases the product is odd. The exterior product is commutative in the graded sense and associative.
Let {eκ}κ=1,... ,m be a base of V and let {e
κ}κ=1,... ,m be the dual base. Each element e
κ defines an
even covector
eκe :V × O(V ) → R: (v, o) 7→ 〈e
κ, v〉. (4)
We choose an orientation o of V and introduce odd 0-covector eo defined by eo(o) = −eo(Po) = 1 and
the even covector ee defined by ee(o) = ee(Po) = 1.
We come now to multivectors. We denote by K(×qV × O(V )) the vector space of formal linear
combinations of sequences (v1, v2, . . . , vq, o) ∈ ×
qV ×O(V ). In the space K(×qV ×O(V )) we introduce
subspaces
A pq (V ) =
i=1 λi(v
2, . . . , v
i) ∈ K(×qV × O(V ));
i=1 λia(v
2, . . . , v
i) = 0 for each a ∈ ∧qpV
. (5)
Subsequently we define quotient spaces ∧qpV = K(×
qV × O(V ))
A pq (V ). Elements of spaces ∧
eV and
∧qoV are called even q - vectors and odd q - vectors respectively. A multivector is said to be simple if
it is represented by a single element of the space V q × O(V ) interpreted as a subspace of K(V q ×
O(V )). Evaluation of q - covectors on sequences (v1, v2, . . . , vq, o) ∈ ×
qV × O(V ) extends to linear
combinations and their equivalence classes. If w is a q - vector represented by the linear combination∑n
i=1 λi(v
2, . . . , v
i) and a is a q - covector of the same parity as w, then
〈a, w〉 =
i=1 λia(v
2, . . . , v
i) (6)
is the evaluation of a on w. We have constructed pairings 〈 , 〉:∧qpV
∗×∧qpV → R. The exterior product
of multivectors can be easily defined using representatives. The parity of the exterior product is odd
if the parity of one of the factors is odd. It is even otherwise. The exterior product is commutative in
the graded sense and associative.
The left interior multiplications are the operations :∧qpV × ∧
V ∗ → ∧
V ∗, defined for q 6 q′
by 〈w a,w′〉 = 〈a, w ∧w′〉. The parity pp′ which appears in this definition is constructed by assigning
the numerical values +1 and −1 to e and p respectively. The parity of the multivector w′ must
match the parity of the multicovector w a. The right interior multiplications are the operations
:∧qpV×∧
∗ → ∧
pp′ V, defined for q > q
′ by 〈a′, w a〉 = 〈a′∧a, w〉. The parity of the multicovector
a′ in this definition must match the parity of the multivector w a.
3.The Weyl isomorphism and a useful formula.
The space ∧mo V
∗ is one-dimensional. This makes it possible to define the tensor product ∧qeV ⊗∧
as the set of equivalence classes of pairs (w, e) ∈ ∧qeV ×∧
∗. Pairs (w, e) and (w′, e′) are equivalent
if there is a number λ such that w′ = λw and e = λe′ or w = λw′ and e′ = λe. The equivalence class
of a pair (w, e) will be denoted by w ⊗ e. A tensor a ∈ ∧qeV ⊗ ∧
∗ will always be presented as a
product w ⊗ e. The set ∧qeV ⊗ ∧
∗ is a vector space (see [5]).
Proposition 1 ([5]). The linear mapping
Weq:∧
eV ⊗ ∧
∗ → ∧m−qo V
∗:w ⊗ e 7→ w e (7)
is an isomorphism.
The mapping Weq is called the Weyl isomorphism.
We will show now that the values of any bilinear mapping b:∧qeV
∗ × ∧q
∗ → ∧mo V
∗ can be
expressed using the exterior product. The following three technical lemmas will be used.
Lemma 1. Let w ∈ ∧qeV . Then
w (eo ∧ e
e ∧ . . . ∧ e
e ) =
ν1<...<νq
(νi−i)〈eν1e ∧ . . . ∧ e
e , w〉eo ∧ e
e ∧ . . . ∧ e
e (8)
where νq+1 < . . . < νm and (νq+1, . . . , νm) denotes the complementary (m− q)-tuple of (ν1, . . . , νq).
Proof: It is enough to prove the claim for a simple q-vector. If w = w1 ∧ . . . ∧ wq, then
(w1 ∧ . . . ∧ wq) (eo ∧ e
e ∧ . . . ∧ e
(−1)ν1−1〈eν1e , w1〉wq
. . .
eo ∧ e
e ∧ . . . ∧ ê
e ∧ . . . ∧ e
= . . . =
ν1 6=...6=νq
(νi−i) sgn(ν1, . . . , νq)〈e
e , w1〉 · · · 〈e
e , wq〉
eo ∧ e
1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê
e ∧ . . . ∧ e
ν1<...<νq
σ∈S(q)
(νi−i) sgnσ〈e
νσ(1)
e , w1〉 · · · 〈e
νσ(q)
e , wq〉
eo ∧ e
1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê
e ∧ . . . ∧ e
ν1<...<νq
σ∈S(q)
(νi−i) sgnσ〈e
νσ(1)
e , w1〉 · · · 〈e
νσ(q)
e , wq〉eo ∧ e
νq+1 ∧ . . . ∧ eνme
ν1<...<νq
(νi−i) det(〈eνre , ws〉)eo ∧ e
νq+1 ∧ . . . ∧ eνme . (9)
In the second line of this sequence of equalities we used the well known identity
v (a1 ∧ . . . ∧ aq) =
(−1)ν−1〈aν , v〉a1 ∧ . . . ∧ âν ∧ . . . ∧ aq (10)
where v is a vector and a1, . . . , aq are covectors. In the third line we applied repeatedly this identity
and we noted that when ν1 < . . . < νq each missing covector e
e in the expression eo∧e
1∧. . .∧êν1e ∧. . .∧
e ∧ . . .∧ e
e occupied the (νi− i+ 1)-th place in the exterior product, otherwise if νi−l−1 < νi < νi−l,
then it occupied the (νi − i − l + 1)-th place, hence we get the factor (−1)
(νi−i) sgn(ν1, . . . , νq)
where the symbol sgn(ν1, . . . , νq) denotes the sign of the permutation (ν1, . . . , νq).
Lemma 2. If a ∈ ∧qeV
∗ and w ⊗ e ∈ ∧qeV ⊗ ∧
∗, then
〈a, w〉e = a ∧ Weq(w ⊗ e). (11)
Proof: It is enough to prove the claim for e = eo ∧ e
e ∧ . . . ∧ e
e . By the Lemma 1 we get
a ∧ Weq(w ⊗ e)
= a ∧ (w (eo ∧ e
e ∧ . . . ∧ e
ν1<...<νq
(νi−i)〈w, eν1e ∧ . . . ∧ e
e 〉a ∧ eo ∧ e
e ∧ . . . ∧ e
ν1<...<νq
(νi−i)w
ν1...νq
µ1<...<µq
aµ1...µqee ∧ e
e ∧ . . . ∧ e
e ∧ eo ∧ e
e ∧ . . . ∧ e
e . (12)
The right-hand side of (12) reduces to
ν1<...<νq
(νi−i)wν1...νqaν1...νqeo ∧ e
e ∧ . . . ∧ e
e , (13)
since νq+1 < . . . < νm and (νq+1, . . . , νm) is the complementary (m− q)-tuple of (ν1, . . . , νq). Finally
we note that moving each νi (for i = 1, . . . , q) to the νi-th place in (12) requires νi − i transpositions,
since ν1 < . . . < νq. Then each of the remaining νi, i.e. those with i = q + 1, . . . ,m, will necessarily
be at the νi-th place. Hence we obtain
a ∧ Weq(w ⊗ e) =
ν1<...<νq
aν1...νqw
ν1...νqeo ∧ e
e ∧ . . . ∧ e
e . (14)
We denote by Hom(∧mo V
∗| ∧qe V
∗) the space of linear mappings from ∧qeV
∗ to ∧mo V
∗ and by
iq: Hom(∧
∗| ∧qe V
∗) → ∧qeV ⊗ ∧
∗ (15)
the isomorphism characterized by
〈iq(l), a
′ ⊗ u〉 = 〈l(a′), u〉 (16)
for each l ∈ Hom(∧mo V
∗| ∧qe V
∗), a′ ∈ ∧qeV
∗ and u ∈ ∧mo V . The pairing
〈 , 〉: (∧qeV ⊗ ∧
∗) × (∧qeV
∗ ⊗ ∧mo V ) → R: (w ⊗ e, a⊗ u) 7→ 〈a, w〉〈e, u〉 (17)
is used.
Lemma 3. If l ∈ Hom(∧mo V
∗| ∧qe V
∗), then
l(a) = a ∧ Weq(iq(l)), (18)
for each a ∈ ∧qeV
Proof: Let iq(l) = wl ⊗ e with wl ∈ ∧
eV . Using Lemma 2 we obtain
a ∧ Weq(i(l)) = a ∧ Weq(wl ⊗ e) = 〈a, wl〉e. (19)
We will show now that l(a) = 〈a, wl〉e. Indeed if we denote by u ∈ ∧
o V the dual basis of e then we
have l(a) = 〈l(a), u〉e and
〈l(a), u〉 = 〈i(l), a⊗ u〉 = 〈wl ⊗ e, a⊗ u〉 = 〈a, wl〉〈e, u〉 = 〈a, wl〉. (20)
We can now finally show that a useful expression can be obtained for the values of any bilinear
mapping
b:∧qeV
∗ × ∧q
∗ → ∧mo V
∗. (21)
We associate with b the linear mappings
b:∧qeV
∗ → ∧q
e V ⊗ ∧
∗: a 7→ iq′(b(a, ·)) (22)
∗ → ∧qeV ⊗ ∧
∗: a′ 7→ iq(b(·, a
′)). (23)
Proposition 2. If b as in (21) is a bilinear mapping and b, b are the associated linear mappings (22)
and (23), then
b(a, a′) = a′ ∧ Weq′(b(a)) = a ∧ Weq(b(a
′)). (24)
for each (a, a′) ∈ ∧qeV
∗ × ∧q
Proof: Applying Lemma 3 to l = b(a, ·) we get
b(a, a′) = b(a, ·)(a′) = a′ ∧ Weq′(iq′(b(a, ·))) = a
′ ∧ Weq′(b(a)). (25)
The other equality is obtained in the same way by applying Lemma 3 to l = b(·, a′).
4. Integration of differential forms. Chains and currents.
Let M be an affine space modelled on a vector space V . A differential q-form on M is a differentiable
function A:M ××qV × O(V ) → R depending on a point, q vectors and an orientation. It is q-linear
and totally antisymmetric in its vector arguments. A differential form A is said to be even or odd if
for each point in x ∈ M it defines a even or odd multicovector, respectively. We note that a zero-form
on M is a differentiable function f :M ×O(V ) → R. The vector space of even differential q-forms will
be denoted by Φqe(M) and space of odd differential q-forms will be denoted by Φ
o(M). The symbol
Φqp(M) will be used to denote either of the two spaces when the distinction is of no importance.
For the definition of the exterior product of two differential forms and for that of exterior differential
of a differential form, we refer the reader to [5]. Here we just recall that if both forms A and A′ are
even or both are odd, the product A ∧ A′ is even. In other cases the product is odd. The parity of
the differential dA of a q-form A is the same as the parity of the original form A. A q-form A can
be interpreted as a q-covector field Ã:M → ∧qpV
∗. The exterior product and the exterior differential
are extended to this alternative interpretation of forms. The left and right interior multiplications of
even and odd multivector fields with even and odd multicovector fields are defined point by point in
an obvious manner.
A cell of dimension q or a q - cell in M is a pair (χ, o), where χ is a differentiable mapping χ:Rq → M
and o is an orientation of V . For q = 0, R0 is the vector space {0} with a single element 0. Hence a
zero-cell in M is a pair of a point x ∈ M and an orientation of V . The integral of a q - form A on a
cell (χ, o) is the Riemann integral
(χ,o)
· · ·
A (χ(s1, . . . , sq),D1χ(s1, . . . , sq), . . . ,Dqχ(s1, . . . , sq), o) ds1 · · · dsq. (26)
The integral of a 0 - form f on a zero-cell (x, o) is the value
(x,o)
f = f(x, o). For each q we introduce
the space K(Xq(M)) of formal linear combinations of q - cells. The formal linear combinations turn
into real linear combinations if cells are identified with elements of K(Xq(M)). Integration of forms is
extended to linear combinations by linearity. Subspaces N pq (M) ⊂ K(Xq(M)) are defined as the sets
q (M) =
C ∈ K(Xq(M));
A = 0 for each A ∈ Φqp(M)
. (27)
Elements of the quotient spaces Ξpq(M) = K(Xq(M))
N pq (M) are called even chains or odd chains of
dimension q. We extend the sequence of even and odd chains to negative dimension q by defining the
spaces Ξpq(M) = {0} for each q < 0. A chain is said to be simple if it has a single cell as one of its
representatives. Integrals of q - forms on q - chains are well defined. The integral of a q - form A on the
class C of C ∈ K(Xq(M)) is the integral of A on C.
The boundary operator ∂ assigns to a chain C ∈ Ξ pq (M) its boundary ∂C ∈ Ξ
q−1(M). The boundary
of a simple chain represented by a q - cell (χ, o) is the chain represented by the combination
i=1(−1)
i−1((χ(i,1), o) − (χ(i,0), o)), (28)
where the (q−1) - cells (χ(i,1), o) and (χ(i,0), o) are defined by
(i,∗):Rq−1 → M : (s1, . . . , ŝi, . . . , sq) 7→ χ(s1, . . . , si−1, ∗, si+1 . . . , sq) (29)
with ∗ replaced by 1 and 0, respectively. The cells introduced in (29) represent the faces of the simple
chain. The construction of the boundary is extended to generic chains by linearity. The boundary of
a boundary is the zero chain. It is known that Stokes theorem holds for chains and forms of the same
parity.
An even or odd de Rham current of dimension q on a manifold M is a linear function c: Φqp(M) → R.
We will use the symbol
A to denote the value c(A). The spaces of forms are given certain topologies
and the continuity of the function is required. Chains will be treated as currents. They form a dense
subspace in the space of currents. We will consider only very simple examples of currents other than
chains. Topological considerations are of little importance for these examples. The boundary of a
current is defined by assuming that Stokes theorem holds for currents. Thus if c is a current of
dimension q, then the boundary of c is the mapping
∂c: Φq−1p (M) → R:A 7→
dA. (30)
In addition to chains the odd de Rham current most frequently used is the Dirac current wδ(x) of
dimension m derived from an odd m - vector w and a point x ∈ M . If A is an odd m - form, then
wδ(x)
A = 〈Ã(x), w〉. (31)
B. Electrodynamics
1.The space of fields.
In this section we will construct a suitable space of fields for electrodynamics which has the same
role of the space of motions in mechanics (see [3]). The space of fields is not a differentiable manifolds.
Nevertheless, we will introduce in this space enough structure to permit the construction of tangent
and cotangent vectors which are the objects needed in in the formulation of the variational principle.
A class of admissible functions defined on the space of fields has also to be specified. This class needs
to be large enough to include the action, which for electrodynamics is generated (in a sense that will be
specified) from a quadratic Lagrangian density, due to the linearity of the theory. Thus, only functions
generated from quadratic mappings will be used in our variational formulation of electrodynamics. A
larger class of admissible functions would be needed to deal with more general, non linear field theories,
and the price of increased technical difficulties should be paid.
Let M be the affine Minkowski space-time of special relativity with the 4-dimensional model space
V and the non degenerate metric tensor g : V → V ∗ of signature (1, 3). The space of odd 4-currents
with compact supports in M will be denoted by CM . Differential forms will always be presented as
covector fields.
We consider the set X(Φ1e(M);CM) of pairs (A, c), where c is an odd current of dimension 4 in M
with a compact support Sup(c) and A is a local even 1 - form
A:U → ∧1eV
∗ (32)
defined in an open set U ⊂ M containing the support of c. The 1 - form A will represent the electro-
magnetic potential.
A mapping
κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗ (33)
is said to be quadratic if for each x ∈ M there exists a symmetric bilinear mapping
∗ × ∧2eV
∗ × ∧2eV
∗) → ∧4oV ∗ (34)
such that the mappings κx = κ(x, ·, ·) and δ
2κx satisfy the equality
κx(a, f) =
κx((a, f)(a, f)), (35)
for each (a, f) ∈ ∧1eV
∗ × ∧2eV
∗. We are using the standard definition of quadratic mappings in terms
of polarizations. The polarization of the mapping κx is the mapping δ
2κx and the equation (35) is the
standard relation between a quadratic mapping defined on a vector space and its polarization. We will
use the set of quadratic mappings (33) to introduce an equivalence relation in the set X(Φ1e(M);CM).
In the next section they will be also used to define the admissible functions defined on the space of
fields.
Pairs (A, c) and (A′, c′) are equivalent if
κ ◦ (x,A′, dA′) =
κ ◦ (x,A, dA) (36)
for each quadratic mapping κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗. The symbol x is used to indicate the
identity mapping of M and also a point of M .
If µ is an arbitrary odd 4-form on M and κ is set to be the mapping
κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (x, a, f) 7→ µ(x), (37)
then the equivalence condition (36) reduces to
µ (38)
and implies that c′ = c.
Equivalence classes of elements of X(Φ1e(M);CM) will be called fields. Our fields are similar to
those used by Freed in [14]. The space of fields will be denoted by Q(Φ1e(M);CM) or simply Q. The
equivalence class of (A, c) will be denoted by q(A, c) or simply q. There is a natural projection
ε:Q(Φ1e(M);CM) → CM : q(A, c) 7→ c (39)
from the space of fields to the space CM of currents in M . It is not difficult to check that each fibre
ε−1(c) of the projection ε is a vector space which will be denoted by the symbol Q(Φ1e(M); c) or Qc.
Indeed, the sum of two pairs (A, c) and (A′, c) with A:U → ∧1eV
∗ and A′:U ′ → ∧1eV
∗ is the pair
(A + A′, c) defined in U ∩ U ′ ⊃ Sup(c). Descending to the quotient with respect to the equivalence
relation defined by (36) easily gives the sum in Qc. Therefore, the space of fields is the disjoint union
of vector spaces Q =
2.Functions, vertical tangent vectors and covectors in the space of fields.
With each quadratic mapping κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗ we associate the function
k:Q(Φ1e(M);CM) → R: q(A, c) 7→
κ ◦ (x,A, dA). (40)
The above mentioned admissible functions defined on the space of fields are those constructed in this
way. They will be called differentiable. The space of such functions will be denoted by K(Φ1e(M);CM).
We note that from the definition of the space of fields Q = Q(Φ1e(M);CM) (as a quotient space with
respect to the equivalence condition (36)) follows easily that the differentiable functions separate points
of Q, i.e. if k(q′) = k(q) for each k ∈ K(Φ1e(M);CM), then q
′ = q. Indeed if k(q(A′, c′)) = k(q(A, c)) for
each k ∈ K(Φ1e(M);CM), then equation (36) holds for each quadratic mapping κ:M×∧
∗×∧2eV
∗. It follows that c′ = c and q(A′, c) = q(A, c).
The tangent space to a vector space Qc (i.e. to a fibre of ε) coincides with Qc.
The vertical tangent bundle to the vector bundle Q(Φ1e(M);CM) is the space
VQ = Q ×
(ε,ε)
Q = {(q, δq) ∈ Q×Q; ε(q) = ε(δq)} (41)
where q = q(A, c) and δq = q(δA, c) and A:U → ∧1eV
∗, δA:U → ∧1eV
∗. The tangent projection is the
canonical projection
τQ:VQ → Q: (q, δq) 7→ q. (42)
There is no obvious choice of the bundle dual to VQ.
We will use the fibre derivatives of functions k ∈ K(Φ1e(M);CM) as models of covectors. The
derivative Dk of a function k:Q → R is defined for each current c separately. It is evaluated on a pair
of vectors q = q(A, c) ∈ Qc and δq = q(δA, c) ∈ Qc. The result is the expression
Dk (q(A, c), q(δA, c)) =
k (q(A + sδA, c)) (0)
κ ◦ (x,A + sδA, F + sδF )
, (43)
where F = dA and δF = dδA. The symbol dk(q(A, c)) will be used to denote the covector characterized
by the pairing
〈dk (q (A, c)) , q(δA, c)〉 = Dk (q(A, c), q(δA, c)) (44)
with vectors δq = q(δA, c) ∈ Qc.
So we need to calculate the expression
κ ◦ (x,A + sδA, F + sδF )
(x) = Dκx(A(x), F (x), δA(x) ⊕ δF (x)), (45)
for each x ∈ U . We are using the following known definition. If V1, V2,W are vector spaces and
m:V1 × V2 → W is a smooth mapping, then its derivative is the mapping
Dm:V1 × V2 × (V1 ⊕ V2) → W : (a, f, δa⊕ δf) 7→
(m(a + sδa, f + sδf)) (0). (46)
We define for each x ∈ U the bilinear mappings
∗ × ∧1eV
∗ → ∧4oV
∗: (a, a′) 7→ δ2κx((a, 0), (a
, 0)), (47)
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ δ2κx((a, 0), (0, f)) (48)
∗ × ∧2eV
∗ → ∧4oV
∗: (f, f ′) 7→ δ2κx((0, f), (0, f
′)), (49)
obtaining the equality
κx(a, f) =
λx(a, a) + µx(a, f) +
νx(f, f), (50)
for each (a, f) ∈ ∧1eV
∗ × ∧2eV
∗. The equality (50) in terms of the bilinear mappings (47), (48), and
(49) will be useful to calculate the expression (45).
We have the following lemmas.
Lemma 4. If κx:∧
∗ × ∧2eV
∗ → ∧4oV
∗ is a quadratic mapping and λx, µx, νx are the mappings
defined above, then
Dκx(a, f, δa⊕ δf) =
Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) +
Dνx(f, f, δf ⊕ δf), (51)
for each (a, f, δa⊕ δf) ∈ ∧1eV
∗ × ∧2eV
∗ × ∧1eV
∗ ⊕ ∧2eV
Proof: We have the equality
λx ◦ ∆ ◦ pr1 + µx +
νx ◦ ∆
′ ◦ pr2, (52)
where
pr1:∧
∗ × ∧2eV
∗ → ∧1eV
∗: (a, f) 7→ a, (53)
∆:∧1eV
∗ → ∧1eV
∗ × ∧1eV
∗: a 7→ (a, a), (54)
pr2:∧
∗ × ∧2eV
∗ → ∧2eV
∗: (a, f) 7→ f, (55)
∆′:∧2eV
∗ → ∧2eV
∗ × ∧2eV
∗: a 7→ (a, a). (56)
The claim follows from the equality (52) by noting that
Dκx(a, f, δa⊕ δf) =
Dλx ◦ (∆,D∆) ◦ (pr1,Dpr1)(a, f, δa⊕ δf) + Dµx(a, f, δa⊕ δf)
Dνx ◦ (∆
,D∆′) ◦ (pr2,Dpr2)(a, f, δa⊕ δf)
Dλx ◦ (∆,D∆)(a, δa) + Dµx(a, f, δa⊕ δf)
Dνx ◦ (∆
′,D∆′)(f, δf)
Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) +
Dνx(f, f, δf ⊕ δf). (57)
Lemma 5. If V1, V2,W are vector spaces and
b:V1 × V2 → W (58)
is a bilinear mapping, then
Db(a, f, δa⊕ δf) = b(a, δf) + b(δa, f), (59)
for each (a, f, δa⊕ δf) ∈ V1 × V2 × (V1 ⊕ V2).
Proof: From the definition (46) of the derivative it follows that
Db(a, f, δa⊕ δf) =
(b(a, f + sδf) + sb(δa, f + sδf)) (0)
b(a, f) + sb(a, δf) + sb(δa, f) + s2b(δa, δf)
= b(a, δf) + b(δa, f). (60)
In view of Lemmas 4 and 5 the integrand in right hand side of the equality (43) reduces to
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) =
Dλx(A(x), A(x), δA(x) ⊕ δA(x))
+ Dµx(A(x), F (x), δA(x) ⊕ δF (x))
Dνx(F (x), F (x), δF (x) ⊕ δF (x))
(λx(A(x), δA(x)) + λx(δA(x), A(x)))
+ µx(A(x), δF (x)) + µx(δA(x), F (x))
(νx(F (x), δF (x)) + νx(δF (x), F (x))) , (61)
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) = λx(A(x), δA(x)) + µx(A(x), δF (x))
+ µx(δA(x), F (x)) + νx(F (x), δF (x)), (62)
since λx and νx are symmetric. By using the Proposition 2 contained in Section 6, this expression
reduces to
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) =δA(x) ∧ We1(λx(A(x)) + δF (x) ∧ We2(µx(A(x))
+ δA(x) ∧ We1(µx(F (x)) + δF (x) ∧ We2(νx(F (x))
=δA(x) ∧
We1(λx(A(x)) + µx(F (x)))
+ δF (x) ∧ (We2(µx(A(x)) + νx(F (x)))) . (63)
where the linear mappings λx and λx are obtained from the bilinear mapping λx as prescribed in
Section 6, i.e.,
∗ → ∧1eV ⊗ ∧
∗: a 7→ i1(λx(a, ·)), (64)
∗ → ∧1eV ⊗ ∧
∗: a 7→ i1(λx(·, a)), (65)
for each x ∈ U . The mappings µx, µx are obtained from µx, and νx, νx are obtained from νx in the
same way.
We define from λx, λx the mappings
λ:M × ∧1eV
∗ → ∧1eV ⊗ ∧
∗: (x, a) 7→ λx(a), (66)
λ:M × ∧1eV
∗ → ∧1eV ⊗ ∧
∗: (x, a) 7→ λx(a). (67)
The mappings µ, µ, ν, ν are defined respectively from µx, µx, νx, νx in the same way.
By using the equalities (47) and (63) which hold for each x ∈ U we obtain the equality
κ ◦ (x,A + sδA, F + sδF )
We1 ◦
λ ◦ (x,A) + µ ◦ (x, F )
+ (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (68)
We have also used the graded commutativity of the exterior product.
Hence,
Dk (q(A, c), q(δA, c)) =
κ ◦ (x,A + sδA, F + sδF )
We1 ◦
λ ◦ (x,A) + µ ◦ (x, F )
(We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (69)
Since F = dA and δF = dδA, this equality can be converted to the form
Dk (q(A, c), q(δA, c)) = −
We1 ◦
λ ◦ (x,A) + µ ◦ (x, dA)
+ d (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, dA)))) ∧ δA
d ((We2 ◦ (µ ◦ (x,A) + ν ◦ dA)) ∧ δA) . (70)
We note that the first integral contains the exterior product of δA and an odd 3-form, while the second
integral contains the exterior differential of the product of δA and an odd 2-form.
The obtained expression suggests the representation of covectors as equivalence classes of elements
of the sets X(Φ2o(M) × Φ
o(M);CM) introduced below.
An element of X(Φ2o(M)×Φ
o(M);CM) is a triple (G, J, c) of a local odd differential 2-form G:U →
∗, a local odd differential 3-form J :U → ∧3oV
∗ and a current c with support contained in U . The
odd 2-form G will be interpreted as the electromagnetic induction, while the odd 3-form J will be
interpreted as the current. A covector p is an equivalence class p(G, J, c) of (G, J, c) ∈ X(Φ2o(M) ×
Φ3o(M);CM). The equivalence relations in X(Φ
o(M) × Φ
o(M);CM) are based on the expression
J ∧ δA−
d (G ∧ δA)
. (71)
Elements (G, J, c) and (G′, J ′, c′) are equivalent if c = c′ and
J ′ ∧ δA−
d (G′ ∧ δA)
J ∧ δA−
d (G ∧ δA)
, (72)
for each (δA, c) ∈ Qc.
This equivalence relation is related to the equivalence relation (36). They can be used in the
construction of dual objects. Indeed a (vertical) tangent vector might also (more generally) be defined
as an equivalence class of pairs (δA, c) with respect to an equivalence relation similar to (36). A
convenient representation of such relation is a relation similar to (72), but with (G, J, c) fixed, defining
the equivalence between two pairs (δA, c) and (δA′, c). Such a construction is not needed in our case
since each Qc is a vector space, so that its tangent space is canonically identified with Qc itself.
The vector space Πc of covectors associated with the current c will be used as the dual space of the
space Qc thought of as the space of vertical tangent vectors with the pairing
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
, (73)
where δq = q(δA, c) ∈ Qc and p = p(G, J, c) ∈ Πc. The space of all covectors is the union
Πc. (74)
There is a natural projection
ε′: Π → CM : p(G, J, c) 7→ c. (75)
from the space of fields to the space CM of currents in M . The phase space is the space
Ph = Q ×
(ε,ε′)
Π = {(q, p) ∈ Q× Π; ε(q) = ε′(a)}. (76)
The symbol Phc will denote the set Qc × Πc ⊂ Ph.
3.A virtual action principle for electrodynamics.
In this section a variational principle for electrodynamics similar to the virtual action principle of
analytical mechanics (see [3]) will be formulated. The construction is an example of similar construc-
tions needed for the variational formulation of a general field theory. The linearity of the theory and
the choice of formulating it on the affine Minkowski space makes our presentation simpler though
containing the core of the general framework (which can be found in [1]).
The action is the differentiable function
W :Q(Φ1e(M);CM) → R: q(A, c) 7→
L ◦ (A, dA) (77)
derived from the quadratic Lagrangian density
L:∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ −
〈f,∧2eg
−1(f)〉
|g|. (78)
We are denoting by
|g| the odd 4-covector defined by the Minkowski metric (see [5]). Later we
will use the same symbol to denote the constant 4-covector field constructed from it.
The 1-form A is called the potential, the 2-form F = dA is called the electromagnetic field.
We note that the Lagrangian is a quadratic mapping which depends only on its second argument f
and thus λ = µ = 0 and ν does not depend on x in the formula (50).
A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if the equality
〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉
= 0 (79)
holds for each virtual displacement δq = q(δA, c) ∈ Qc. For each current c the dynamics associated
with the current c is the set Dc ⊂ Phc of phases which satisfy the virtual action principle. The
dynamics is the subset
Dc (80)
of the phase space Ph.
A phase space trajectory is a triple of local differential forms
(A,G, J):U → ∧1eV
∗ × ∧2oV
∗ × ∧3oV
. (81)
The dynamics of a system can also be represented as a set DDDD of phase space trajectories (A,G, J) such
that for each current c with support included in U the phase ph = (q(A, c), p(G, J, c)) is in Dc.
The equation (79) is too abstract to be used directly. A more concrete expression is given in the
following proposition.
Proposition 3. A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if and only if
the equality
−1 ◦ dA
∧ δA− d
−1 ◦ dA
J ∧ δA−
d (G ∧ δA)
is satisfied for each virtual displacement q(δA, c).
Proof: By applying the formula (70) to the action W we obtain that its variation is
〈dW (q), δq〉 =
(−d(We2 ◦ ν ◦ dA) ∧ δA + d ((We2 ◦ ν ◦ dA) ∧ δA)) , (83)
since λ = µ = 0. Moreover
νx(f, f) = −
〈f,∧2eg
−1(f)〉
|g|, (84)
from which it follows after some calculation that
ν(x, f) = −
∧2e g
−1(f) ⊗
|g|, (85)
for each f ∈ ∧2eV
∗. Hence,
We2 ◦ ν ◦ (x, dA) = −
−1 ◦ dA
and the variation of the action reduces to
〈dW (q), δq〉 =
−1 ◦ dA
∧ δA− d
−1 ◦ dA
. (87)
Recalling that, on the other hand
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
, (88)
the claim follows.
A phase space trajectory belongs to the dynamics DDDD, if and only if it satisfies the virtual action
principle for each current c with support included in its domain of definition. There is a characterization
of the dynamics of phase space trajectories in terms of differential equations. This is shown in the
following propositions.
Theorem 4. A phase space trajectory (A,G, J) belongs to the dynamics DDDD if and only if it satisfies
the Euler-Lagrange equation
−1 ◦ dA
J (89)
and the constitutive relation
−1 ◦ dA
|g|. (90)
Proof: If a phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the constitutive
relation, then by substituting the expressions (89) and (90) of J and G in terms of the electromagnetic
field F in the virtual action principle (82) it follows that (A,G, J) belongs to the dynamics DDDD.
The inverse implication will be proved in the next section.
The constitutive relation (90) produced by our variational principle corresponds to the momentum-
velocity relation of analytical mechanics.
Proposition 5. A phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the
constitutive relation if and only if it satisfies the Maxwell’s equations
J (91)
and the constitutive relation
−1 ◦ F
|g|, (92)
with F = dA.
Proof: The constitutive relation (91) is satisfied if and only if the equation (90) is satisfied for
F = dA.
If a phase space trajectory (A,G, J) satisfies the Maxwell’s equations and the constitutive relation,
then by substituting the expression (92) of the electromagnetic induction in the equation (91) we see
that the Euler-Lagrange equation is satisfied, since F = dA.
Conversely if the Euler-Lagrange equation and the constitutive relation are satisfied, then, again by
substitution, we obtain that the Maxwell’s equations (91) are satisfied.
We remark that since the virtual action principle (77) is more complete than the Hamilton principle,
our formulation leads to the Maxwell’s equations (91) with external sources. The proposed variational
principle also permits the derivation of the constitutive relations (92) which are usually postulated
separately since the variations normally considered are not general enough to derive them from the
variational principle.
4.The Dynamics in a compact domain.
Let the current c consist in integrating an odd 4-form on a compact domain K ⊂ M with smooth
boundary ∂K. Field configurations, tangent vectors and covectors are equivalence classes of equivalence
relations based on the expressions ∫
κ ◦ (x,A, dA) (93)
and ∫
J ∧ δA−
d (G ∧ δA)
J ∧ δA−
G ∧ δA. (94)
It follows that a field q = q(A,K) is represented by the restriction
A|K:K → ∧1eV
∗ (95)
of the potential A to the the domain K. A tangent vector δq = q(δA,K) is represented by the
restriction
(δA)|K:K → ∧1eV
∗ (96)
of the variation δA to the domain K. A covector p = p(G, J,K) is represented by the pair of the
restriction
G|∂K: ∂K → ∧2oV
∗ (97)
of the electromagnetic induction G to the boundary ∂K of the domain K and the restriction
K → ∧3oV
∗ (98)
of the current J to the interior
K of the domain K.
A phase is a pair (q, p) of a field q = q(A,K) and a covector p = p(G, J,K).
The action is
W :QK → R: q(A,K) 7→
L ◦ (A, dA). (99)
The virtual action principle is the equality
〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉
= 0 (100)
and the dynamics in the domain K is the set DK ⊂ Ph of phases satisfying the virtual action principle.
Proposition 6. A phase ph = (q(A,K), p(G, J,K)), defined in a compact domain K, belongs to the
dynamics DK if and only if the Euler-Lagrange equation
−1 ◦ dA
K (101)
and the constitutive relation
G|∂K =
−1 ◦ dA
|∂K (102)
are satisfied.
Proof: If q = q(A,K), p = p(G, J,K) and δq = q(δA,K), then
〈dW (q), δq〉 =
−1 ◦ dA
−1 ◦ dA
−1 ◦ dA
−1 ◦ dA
∧ δA. (103)
On the other hand
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
G ∧ δA. (104)
Thus the virtual action principle assumes the form
−1 ◦ dA
∧ δA−
−1 ◦ dA
J ∧ δA−
G ∧ δA. (105)
By using variations with (δA)|∂K = 0 we derive the Euler-Lagrange equation
−1 ◦ dA
K. (106)
Assuming that this equation is satisfied and using arbitrary variations, the constitutive relation
G|∂K =
−1 ◦ dA
|∂K (107)
follows.
The following Proposition completes the proof of the Theorem 4.
Proposition 7. If a phase space trajectory
(A,G, J):U → ∧1eV
∗ × ∧2oV
∗ × ∧3oV
. (108)
belongs to the dynamics DDDD, then the Euler-Lagrange equation
−1 ◦ dA
J (109)
and the constitutive relation
−1 ◦ dA
|g| (110)
are satisfied in U .
Proof: If (A,G, J) is a phase space trajectory, defined in the open set U ⊂ M , which belongs to the
dynamics DDDD, then the equation
−1 ◦ dA
K (111)
and the boundary relation
G|∂K =
−1 ◦ dA
|∂K (112)
are satisfied for each compact domain K ⊂ U . It follows that equations (109) and (110) are satisfied
in every x ∈ U .
5.The Lagrangian formulation of electrodynamics.
The Lagrangian formulation of dynamics is the infinitesimal limit of the formulation in a compact
domain with the domain shrinking to a point. A formal method which greatly simplifies the passage
to the infinitesimal limit is to replace the compact domain — which is used exclusively as domain
of integration — with the current c = δ(x)w, where δ(x) is the Dirac delta function in x ∈ M and
w ∈ ∧4oV is an odd 4-vector, with w 6= 0. The construction of infinitesimal fields, tangent vectors and
covectors is based on the expressions
δ(x)w
κ ◦ (x,A, dA) = 〈κ (x,A(x), dA(x)) , w〉 (113)
δ(x)w
J ∧ δA−
d (G ∧ δA)
J(x) ∧ δA(x) −
d(G ∧ δA)(x), w
. (114)
Since w 6= 0 and ∧4oV is one-dimensional, it follows from the first expression that a field q = q(A, c) is
represented by the pair
(A(x), F (x)) ∈ ∧1eV
∗ × ∧2eV
∗ (115)
of an even 1-covector A(x) and an even 2-covector F (x).
The second expression reduces to
dG(x) −
∧ δA(x) + G(x) ∧ δF (x), w
, (116)
since dδA(x) = δF (x).
It follows that a tangent vector δq = q(δA, c) is represented by the pair
(δA(x), δF (x)) ∈ ∧1eV
∗ × ∧2eV
∗ (117)
and a covector p = p(G, J, c) is represented by the pair
G(x), dG(x) −
∈ ∧2oV
∗ × ∧3oV
∗. (118)
The pairing
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
(119)
assumes the form
〈〈〈〈〈p, δq〉〉〉〉〉
dG(x) −
∧ δA(x) + G(x) ∧ δF (x), w
. (120)
We have constructed the space of infinitesimal fields Qδ = ∧
∗×∧2eV
∗ and the space of infinitesimal
covectors Πδ = ∧
∗ × ∧3oV
∗. Hence, the infinitesimal phase space is
Phδ = Qδ × Πδ = ∧
∗ × ∧2eV
∗ × ∧2oV
∗ × ∧3oV
∗. (121)
The infinitesimal action is
W (q(A, δ(x)w)) = 〈L(A(x), F (x)), w〉 . (122)
The infinitesimal dynamics is the set
(q, δq) ∈ Phδ; ∀δq∈Qδ 〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉
, (123)
It is easy to verify that the infinitesimal dynamics Dδ admits also the following more explicit expression
(a, f, g, h) ∈ Phδ; ∀
(δa,δf)∈∧1eV
∗×∧2eV ∗
DL(a, f, δa, δf) = −
(h ∧ δa + g ∧ δf)
. (124)
The infinitesimal dynamics Dδ is characterized by the following Proposition.
Proposition 8. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to
the infinitesimal dynamics Dδ if and only if the equations
G(x) =
−1(F (x))
|g| (125)
dG(x) =
J(x) (126)
are satisfied.
Proof: If q = q(A, δ(x)w)), p = p(G, J, δ(x)w), and δq = q(δA, δ(x)w), with w 6= 0, then the variation
of the action (87), which can also be expressed in the form
〈dW (q), δq〉 = −
−1 ◦ F
∧ δF, (127)
reduces to
〈dW (q), δq〉 = −
−1(F (x))
∧ δF (x), w
. (128)
Thus the virtual action principle
〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉
(129)
has the explicit form
dG(x) −
∧ δA(x) +
G(x) −
−1(F (x))
∧ δF (x), w
= 0. (130)
Therefore if ph satisfies the equations (125) and (126), then (130) is satisfied and ph ∈ Dδ.
Conversely, if ph ∈ Dδ, then it satisfies the virtual action principle, which implies
dG(x) −
∧ δA(x) +
G(x) −
−1(F (x))
∧ δF (x) = 0, (131)
since w 6= 0 and ∧4oV is one-dimensional. The equations (125) and (126) follow, since δA(x) and δF (x)
are independent.
In a preceding section we showed that the dynamics of phase space trajectories can also be charac-
terized by the Maxwell’s equations and the constitutive relation. This fact can now be proved directly.
Indeed, if a phase space trajectory (A,G, J) belongs to the dynamics DDDD and x is a point in the
domain of definition of the trajectory, then the virtual action principle is satisfied for every infinitesimal
current δ(x)w. Thus the Maxwell’s equations and the constitutive relation are satisfied in x, thanks
to the previous proposition. It follows that they are satisfied in the whole domain of definition of the
trajectory.
To prove the inverse implication, we observe that the virtual action principle (82) can also be
expressed in the form
−1 ◦ F
|g| ∧ δF =
J ∧ δA−
d (G ∧ δA)
(132)
−1 ◦ F
∧ δF +
= 0, (133)
for each virtual displacement q(δA, c). Thus if a phase space trajectory (A,G, J) satisfies the Maxwell’s
equations, then it satisfies the virtual action principle for every current c with support contained in
the domain of definition of the trajectory and hence it belongs to DDDD.
6.The Hamiltonian formulation of electrodynamics.
We associate with the Lagrangian density
L:∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ −
f,∧2eg
−1(f)
|g|, (134)
the energy density
E:∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗ → ∧4oV
∗ (135)
defined by
E(a, g, f) = −
g ∧ f − L(a, f)
g ∧ f +
f,∧2eg
−1(f)
g ∧ f +
−1(f)
2g − ∧2eg
−1(f)
∧ f (136)
and treat this mapping as a family
E(a, g, ·):∧2eV
∗ → ∧4oV
∗ (137)
of mappings on the fibres of the projection
prP :∧
∗ × ∧2oV
∗ × ∧2eV
∗ → ∧1eV
∗ × ∧2oV
∗ (138)
onto the field-momentum space P = ∧1eV
∗ × ∧2oV
∗. The set
Cr(E, prP ) =
(a, g, λ) ∈ ∧1eV
∗ × ∧2oV
∗ × ∧2eV
δλ∈∧2
DE(a, g, λ, 0, 0, δλ) = 0
(139)
is the critical set of the family. The equality
DE(a, g, λ, δa, δg, δλ) = −
(δg ∧ λ + g ∧ δλ) − DL(a, λ, δa, δλ)
(δg ∧ λ + g ∧ δλ) +
−1(λ)
δg ∧ λ +
−1(λ)
(140)
implies
DE(a, g, λ, 0, 0, δλ) = −
−1(λ)
∧ δλ. (141)
Thus we obtain the expression
Cr(E, prP ) =
(a, g, λ) ∈ ∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗; g = ∧2eg
−1(λ)
. (142)
The critical set is the graph of the Legendre mapping
Λ:∧1eV
∗ × ∧2eV
∗ → ∧2oV
∗: (a, λ) 7→ ∧2eg
−1(λ)
|g|. (143)
For each a ∈ ∧1eV
∗, the mapping Λ(a, ·) is invertible. Its inverse is the mapping
Λ(a, ·)−1:∧2oV
∗ → ∧2eV
∗: g 7→ ∧2eg
|g−1| g
, (144)
where
|g−1| denotes the odd 4-vector characterized by 〈
|g−1|〉 = 1. It follows that the critical
set is the image of the section
σ:∧1eV
∗ × ∧2oV
∗ → ∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗: (a, g) 7→
a, g,∧2eg
|g−1| g
(145)
of the projection prP . The Hamiltonian density is the mapping
H = E ◦ σ:∧1eV
∗ × ∧2oV
∗ → ∧4oV
∗, (146)
defined by the formula
H(a, g) = −
|g−1| g
, (147)
for each (a, g) ∈ ∧1eV
∗×∧2oV
∗. The passage from the Lagrangian density L to the Hamiltonian density
H is the Legendre transformation of electrodynamics.
We show that the energy density can be used to generate the infinitesimal dynamics Dδ. We consider
the set
(a, f, g, r) ∈ Phδ; ∃
V ∗ ∀(δa,δg,δλ)∈∧1
V ∗×∧2
V ∗×∧2
DE(a, g, λ, δa, δg, δλ) =
(r ∧ δa− f ∧ δg)
. (148)
This set is obtained by projecting the set
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
(δa,δg,δλ)∈∧1eV
∗×∧2oV ∗×∧2eV ∗
DE(a, g, λ, δa, δg, δλ) =
(r ∧ δa− f ∧ δg)
(149)
onto the phase space Phδ = ∧
∗ × ∧2eV
∗ × ∧2oV
∗ × ∧3oV
It follows from (140) that λ = f and that the set D̃E reduces to
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
∗; λ = f,
(δa,δg,δf)∈∧1eV
∗×∧2oV ∗×∧2eV ∗
DL(a, f, δa, δf) =
(r ∧ δa + g ∧ δf)
. (150)
It projects onto the infinitesimal dynamics
(a, f, g, r) ∈ Phδ; ∀
(δa,δf)∈∧1
V ∗×∧2
DL(a, f, δa, δf) = −
(r ∧ δa + g ∧ δf)
. (151)
Hence, DE = Dδ.
It is clear from the definition of the set D̃E that this set is included in the set
(a, f, g, r, λ) ∈ Phδ × ∧
∗; (a, g, λ) ∈ Cr(E, prP )
. (152)
The use of the mapping σ results in
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
∗; λ = f,
(δa,δg)∈∧1
V ∗×∧2
DH(a, g, δa, δg) =
(r ∧ δa− f ∧ δg)
. (153)
The Hamiltonian description of the dynamics
(a, f, g, r) ∈ Phδ; ∀
(δa,δg)∈∧1
V ∗×∧2
DH(a, g, δa, δg) =
(r ∧ δa− f ∧ δg)
(a, f, g, r) ∈ Phδ; ∧
|g−1| g
= f , r = 0
. (154)
is obtained by projecting onto the phase space Phδ.
Proposition 9. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to
the infinitesimal dynamics Dδ if and only if the equations
F (x) = ∧2eg ◦
|g−1| G(x)
(155)
dG(x) =
J(x) (156)
are satisfied.
Proof: We recall that an infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, is
represented by (
A(x), F (x), dG(x) −
J(x), G(x)
(157)
and use the Hamiltonian description (154) of infinitesimal dynamics Dδ. The claim easily follows.
The resulting equations for the phase space trajectories (A,G, J)
F = ∧2eg ◦
|g−1| G
(158)
J (159)
are called Hamilton’s equations. The equations (159) are again Maxwell’s equations and the equation
(158) is the inverse of constitutive relation (92).
7.References
[1] W. M. Tulczyjew, The origin of variational principles, in the volume Classical and quantum inte-
grability (Warsaw, 2001), 41–75, Banach Center Publ., 59, Polish Acad. Sci., Warsaw, 2003.
[2] G. Marmo, W. M. Tulczyjew, P. Urbański, Dynamics of autonomous systems with external forces,
Acta Physica Polonica B, 33 (2002), 1181–1240.
[3] A. De Nicola, W. M. Tulczyjew, A variational formulation of analytical mechanics in an affine
space, Rep. Math. Phys., 58 (2006), 335–350.
[4] W. M. Tulczyjew, A symplectic framework of linear field theories, Annali di Matematica pura ed
applicata, 130 (1982), 177–195.
[5] G. Marmo, E. Parasecoli, W. M. Tulczyjew, Space-time orientations and Maxwell’s equations, Rep.
Math. Phys. 56 (2005), 209–248.
[6] G. Marmo, W. M. Tulczyjew, Time reflection and the dynamics of particles and antiparticles, Rep.
Math. Phys. 58 (2006), 147–164.
[7] A. De Nicola, W. M. Tulczyjew, A Note on a Variational Formulation of Electrodynamics, Proc. XV
Int. Workshop on Geom. and Phys., Tenerife (Spain), 2006, Publ. de la RSME 11 (2007), 316–323.
[8] J. A. Schouten, Ricci calculus, Springer, Berlin, 1955.
[9] J. A. Schouten, Tensor Analysis for Physicists, Oxford University Press, London, 1951.
[10] G. de Rham, Variétés Differentiables, Hermann, Paris, 1955.
[11] T. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press, Cambridge,
1997.
[12] F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics, Charge, Flux, and
Metric, Birkhäuser, Boston, MA, 2003.
[13] I. V. Lindell, Differential Forms in Electromagnetics, IEEE Press, Piscataway, NJ, and Wiley-
Interscience, 2004.
[14] D. Freed, Classical field theory and Supersimmetry, IAS Park City Mathematics Series Vol. 11,
IAS, Princeton, 2001.

  We present a variational formulation of electrodynamics using de Rham even
and odd differential forms. Our formulation relies on a variational principle
more complete than the Hamilton principle and thus leads to field equations
with external sources and permits the derivation of the constitutive relations.
We interpret a domain in space-time as an odd de Rham 4-current. This permits a
treatment of different types of boundary problems in an unified way. In
particular we obtain a smooth transition to the infinitesimal version by using
a current with a one point support.

Introduction
A general framework for variational formulations of physical theories was presented in [1]. Appli-
cations to statics and dynamics of mechanical systems appear in [2, 3]. In this paper we present a
variational formulation of electrodynamics based on that framework. Our work is related to the gen-
eral formulation of linear field theories in a symplectic framework contained in [4] and to the earlier
formulations of electrodynamics contained in [5, 6]. Some of the results contained in this paper were
announced in [7].
We are presenting a variational formulation of electrodynamics in an intrinsic, frame independent
fashion in the affine Minkowski space-time using de Rham odd and even differential forms ([8, 9, 10])
which permit the rigorous formulations of electrodynamics and the description of the transformation
properties of electromagnetic fields relative to reflections (see [5, 6]). Observable quantities like the
charge contained in a compact volume or the flux of the electromagnetic field through a surface require
the integration of differential forms. The use of odd quantities is not just a matter of elegance. Even
if the space-time is assumed to be orientable, the alternative approach of using standard differential
forms and the Hodge star operator require the choice of a specific orientation, i. e. the addition of an
extrinsic structure. Other recent treatments of classical electrodynamics using odd and even differential
forms can be found in [11, 12, 13].
Our construction is an example of application of the general variational framework [1]. Similar
constructions are needed for the variational formulation of a general field theory. The linearity of
classical electrodynamics and the choice of formulating it on the affine Minkowski space makes our
presentation simpler. Relying on a variational principle more complete than the Hamilton principle
our formulation leads to field equations with external sources. This variational principle also permits
http://arxiv.org/abs/0704.0433v3
the derivation of the constitutive relations which are usually postulated separately since the variations
normally considered are not general enough to derive them from the variational principle.
We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of
different types of boundary problems in an unified way. As an example we obtain a smooth transition to
the infinitesimal version by using a current with a one point support. De Rham currents are essentially
objects dual to differential forms.
The present paper is organized in the following way. In the first part we provide the geometric
structures needed for the rigorous formulation of electrodynamics. Part of this material is based on [5]
and is briefly reported here for the sake of completeness. We recall the Cartan calculus for odd and
even differential forms and their integration theory over odd and even de Rham currents.
The second part contains the main results. We start with the construction of a suitable space of
fields for electrodynamics (not a differential manifold) and a construction of tangent and cotangent
vectors. A convenient representation of these objects is introduced. The definition of the space of
fields is inspired by a similar construction suited for the statics of continuous media which is contained
in the final section of [1]. In Section 3 we formulate a variational principle for electrodynamics similar
to the virtual action principle of analytical mechanics with external forces and boundary terms and
derive the field equations which include the constitutive relations in addition to Maxwell’s equations.
The boundary problem in a finite domain is treated in Section 4. Section 5 contains the Lagrangian
formulation of electrodynamics. The Legendre transformation and the Hamiltonian formulation of
electrodynamics in Section 6 conclude the paper.
A. Preliminaries
Here we provide the geometric structures needed for the rigorous formulation of electrodynamics in
Part B. The material in Sections 1, 2, and 4 is based on [5] and is briefly recalled here for the sake of
completeness. Nevertheless, we add a more explicit presentation of some useful details.
1.Orientations of vector spaces and vector subspaces.
Let V be a vector space of dimension m 6= 0. We denote by F(V ) the space of linear iso-
morphisms from V to Rm called frames. It is known that F(V ) is a homogeneous space with re-
spect to the natural group action of the general linear group GL(m,R) in F(V ). Let GL+(m,R)
and GL−(m,R) be the two connected components of the group GL(m,R). The set of orientations
O(V ) = F(V )
GL+(m,R) has two elements. This set is a homogeneous space for the quotient group
H(m,R) = GL(m,R)
GL+(m,R). The sets E = GL+(m,R) and P = GL−(m,R) are the elements of
the quotient group H(m,R) which is the group of permutations of the two elements of O(V ). There is
an ordered base (e1, e2, . . . , em) of V associated with each frame ξ in a obvious way.
Let W ⊂ V be a subspace of a vector space V . The subspace has the set O(W ) of orientations called
inner orientations of W . Orientations of the quotient space V
W are called outer orientations of W .
An outer orientation o′′ of W can be determined by specifying an inner orientation o of W together
with an orientation o′ of V . Let (e1, . . . , en) be the base of W associated with a frame ξ ∈ o. This base
can be completed to a base (e′1, . . . , e
m) of V with (e
1, . . . , e
n) = (e1, . . . , en). The extended base can
be chosen to be associated with a frame ξ′ ∈ o′. Let π:V → V
W be the canonical projection. The
sequence
(e′′1 , . . . , e
m−n) = (π(e
n+1), . . . , π(e
m)) (1)
is a base of V
W . It determines an orientation o′′ of V
W . Hence an outer orientation of W . The
outer orientation o′′ of W constructed from o ∈ O(W ) and o′ ∈ O(V ) is the same as the orientation
constructed from Po and Po′.
We have introduced inner orientation of subspaces of dimension different from zero and outer orien-
tation of subspaces of codimension different from zero. Integration theory of differential forms requires
the possibility of assigning inner orientations to the subspace W = {0} ⊂ V and outer orientations to
the subspace W = V . Two possible orientations are assigned to the subspace W = {0} ⊂ V one of
which is distinguished. The distinguished orientation is denoted by (+) and the other orientation is
denoted by (−). In agreement with the conventions established for orientation the outer orientations
of the subspace W = {0} ⊂ V are the orientations of V . An outer orientation of W = {0} can be
specified in terms of an inner orientation and an orientation of V . If the inner orientation is (+) and the
orientation of V is o, then o is the outer orientation of W . The orientation Po is the outer orientation
derived from (−) and o. The subspace W = V has a distinguished outer orientation defined as an
orientation of V
2.Multicovectors and multivectors.
A q - covector in a vector space V is a mapping a :×qV ×O(V ) → R. This mapping is q - linear and
totally antisymmetric in its vector arguments. A q - covector a is said to be even, if
a(v1, v2, . . . , vq, Po) = a(v1, v2, . . . , vq, o). (2)
It is said to be odd, if
a(v1, v2, . . . , vq, Po) = −a(v1, v2, . . . , vq, o). (3)
The vector space of even q - covectors will be denoted by ∧qeV
∗ and the space of odd q - covectors will
be denoted by ∧qoV
∗. We will use the symbol ∧qpV
∗ to denote either of the spaces in constructions valid
for both parities. The index p with the two possible values e and o will be used on other occasions.
For the definition of exterior product of (even and odd) multicovectors we refer to [5]. Here we just
recall that if two multicovectors a and a′ are even or both are odd, the product a∧a′ is even. In other
cases the product is odd. The exterior product is commutative in the graded sense and associative.
Let {eκ}κ=1,... ,m be a base of V and let {e
κ}κ=1,... ,m be the dual base. Each element e
κ defines an
even covector
eκe :V × O(V ) → R: (v, o) 7→ 〈e
κ, v〉. (4)
We choose an orientation o of V and introduce odd 0-covector eo defined by eo(o) = −eo(Po) = 1 and
the even covector ee defined by ee(o) = ee(Po) = 1.
We come now to multivectors. We denote by K(×qV × O(V )) the vector space of formal linear
combinations of sequences (v1, v2, . . . , vq, o) ∈ ×
qV ×O(V ). In the space K(×qV ×O(V )) we introduce
subspaces
A pq (V ) =
i=1 λi(v
2, . . . , v
i) ∈ K(×qV × O(V ));
i=1 λia(v
2, . . . , v
i) = 0 for each a ∈ ∧qpV
. (5)
Subsequently we define quotient spaces ∧qpV = K(×
qV × O(V ))
A pq (V ). Elements of spaces ∧
eV and
∧qoV are called even q - vectors and odd q - vectors respectively. A multivector is said to be simple if
it is represented by a single element of the space V q × O(V ) interpreted as a subspace of K(V q ×
O(V )). Evaluation of q - covectors on sequences (v1, v2, . . . , vq, o) ∈ ×
qV × O(V ) extends to linear
combinations and their equivalence classes. If w is a q - vector represented by the linear combination∑n
i=1 λi(v
2, . . . , v
i) and a is a q - covector of the same parity as w, then
〈a, w〉 =
i=1 λia(v
2, . . . , v
i) (6)
is the evaluation of a on w. We have constructed pairings 〈 , 〉:∧qpV
∗×∧qpV → R. The exterior product
of multivectors can be easily defined using representatives. The parity of the exterior product is odd
if the parity of one of the factors is odd. It is even otherwise. The exterior product is commutative in
the graded sense and associative.
The left interior multiplications are the operations :∧qpV × ∧
V ∗ → ∧
V ∗, defined for q 6 q′
by 〈w a,w′〉 = 〈a, w ∧w′〉. The parity pp′ which appears in this definition is constructed by assigning
the numerical values +1 and −1 to e and p respectively. The parity of the multivector w′ must
match the parity of the multicovector w a. The right interior multiplications are the operations
:∧qpV×∧
∗ → ∧
pp′ V, defined for q > q
′ by 〈a′, w a〉 = 〈a′∧a, w〉. The parity of the multicovector
a′ in this definition must match the parity of the multivector w a.
3.The Weyl isomorphism and a useful formula.
The space ∧mo V
∗ is one-dimensional. This makes it possible to define the tensor product ∧qeV ⊗∧
as the set of equivalence classes of pairs (w, e) ∈ ∧qeV ×∧
∗. Pairs (w, e) and (w′, e′) are equivalent
if there is a number λ such that w′ = λw and e = λe′ or w = λw′ and e′ = λe. The equivalence class
of a pair (w, e) will be denoted by w ⊗ e. A tensor a ∈ ∧qeV ⊗ ∧
∗ will always be presented as a
product w ⊗ e. The set ∧qeV ⊗ ∧
∗ is a vector space (see [5]).
Proposition 1 ([5]). The linear mapping
Weq:∧
eV ⊗ ∧
∗ → ∧m−qo V
∗:w ⊗ e 7→ w e (7)
is an isomorphism.
The mapping Weq is called the Weyl isomorphism.
We will show now that the values of any bilinear mapping b:∧qeV
∗ × ∧q
∗ → ∧mo V
∗ can be
expressed using the exterior product. The following three technical lemmas will be used.
Lemma 1. Let w ∈ ∧qeV . Then
w (eo ∧ e
e ∧ . . . ∧ e
e ) =
ν1<...<νq
(νi−i)〈eν1e ∧ . . . ∧ e
e , w〉eo ∧ e
e ∧ . . . ∧ e
e (8)
where νq+1 < . . . < νm and (νq+1, . . . , νm) denotes the complementary (m− q)-tuple of (ν1, . . . , νq).
Proof: It is enough to prove the claim for a simple q-vector. If w = w1 ∧ . . . ∧ wq, then
(w1 ∧ . . . ∧ wq) (eo ∧ e
e ∧ . . . ∧ e
(−1)ν1−1〈eν1e , w1〉wq
. . .
eo ∧ e
e ∧ . . . ∧ ê
e ∧ . . . ∧ e
= . . . =
ν1 6=...6=νq
(νi−i) sgn(ν1, . . . , νq)〈e
e , w1〉 · · · 〈e
e , wq〉
eo ∧ e
1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê
e ∧ . . . ∧ e
ν1<...<νq
σ∈S(q)
(νi−i) sgnσ〈e
νσ(1)
e , w1〉 · · · 〈e
νσ(q)
e , wq〉
eo ∧ e
1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê
e ∧ . . . ∧ e
ν1<...<νq
σ∈S(q)
(νi−i) sgnσ〈e
νσ(1)
e , w1〉 · · · 〈e
νσ(q)
e , wq〉eo ∧ e
νq+1 ∧ . . . ∧ eνme
ν1<...<νq
(νi−i) det(〈eνre , ws〉)eo ∧ e
νq+1 ∧ . . . ∧ eνme . (9)
In the second line of this sequence of equalities we used the well known identity
v (a1 ∧ . . . ∧ aq) =
(−1)ν−1〈aν , v〉a1 ∧ . . . ∧ âν ∧ . . . ∧ aq (10)
where v is a vector and a1, . . . , aq are covectors. In the third line we applied repeatedly this identity
and we noted that when ν1 < . . . < νq each missing covector e
e in the expression eo∧e
1∧. . .∧êν1e ∧. . .∧
e ∧ . . .∧ e
e occupied the (νi− i+ 1)-th place in the exterior product, otherwise if νi−l−1 < νi < νi−l,
then it occupied the (νi − i − l + 1)-th place, hence we get the factor (−1)
(νi−i) sgn(ν1, . . . , νq)
where the symbol sgn(ν1, . . . , νq) denotes the sign of the permutation (ν1, . . . , νq).
Lemma 2. If a ∈ ∧qeV
∗ and w ⊗ e ∈ ∧qeV ⊗ ∧
∗, then
〈a, w〉e = a ∧ Weq(w ⊗ e). (11)
Proof: It is enough to prove the claim for e = eo ∧ e
e ∧ . . . ∧ e
e . By the Lemma 1 we get
a ∧ Weq(w ⊗ e)
= a ∧ (w (eo ∧ e
e ∧ . . . ∧ e
ν1<...<νq
(νi−i)〈w, eν1e ∧ . . . ∧ e
e 〉a ∧ eo ∧ e
e ∧ . . . ∧ e
ν1<...<νq
(νi−i)w
ν1...νq
µ1<...<µq
aµ1...µqee ∧ e
e ∧ . . . ∧ e
e ∧ eo ∧ e
e ∧ . . . ∧ e
e . (12)
The right-hand side of (12) reduces to
ν1<...<νq
(νi−i)wν1...νqaν1...νqeo ∧ e
e ∧ . . . ∧ e
e , (13)
since νq+1 < . . . < νm and (νq+1, . . . , νm) is the complementary (m− q)-tuple of (ν1, . . . , νq). Finally
we note that moving each νi (for i = 1, . . . , q) to the νi-th place in (12) requires νi − i transpositions,
since ν1 < . . . < νq. Then each of the remaining νi, i.e. those with i = q + 1, . . . ,m, will necessarily
be at the νi-th place. Hence we obtain
a ∧ Weq(w ⊗ e) =
ν1<...<νq
aν1...νqw
ν1...νqeo ∧ e
e ∧ . . . ∧ e
e . (14)
We denote by Hom(∧mo V
∗| ∧qe V
∗) the space of linear mappings from ∧qeV
∗ to ∧mo V
∗ and by
iq: Hom(∧
∗| ∧qe V
∗) → ∧qeV ⊗ ∧
∗ (15)
the isomorphism characterized by
〈iq(l), a
′ ⊗ u〉 = 〈l(a′), u〉 (16)
for each l ∈ Hom(∧mo V
∗| ∧qe V
∗), a′ ∈ ∧qeV
∗ and u ∈ ∧mo V . The pairing
〈 , 〉: (∧qeV ⊗ ∧
∗) × (∧qeV
∗ ⊗ ∧mo V ) → R: (w ⊗ e, a⊗ u) 7→ 〈a, w〉〈e, u〉 (17)
is used.
Lemma 3. If l ∈ Hom(∧mo V
∗| ∧qe V
∗), then
l(a) = a ∧ Weq(iq(l)), (18)
for each a ∈ ∧qeV
Proof: Let iq(l) = wl ⊗ e with wl ∈ ∧
eV . Using Lemma 2 we obtain
a ∧ Weq(i(l)) = a ∧ Weq(wl ⊗ e) = 〈a, wl〉e. (19)
We will show now that l(a) = 〈a, wl〉e. Indeed if we denote by u ∈ ∧
o V the dual basis of e then we
have l(a) = 〈l(a), u〉e and
〈l(a), u〉 = 〈i(l), a⊗ u〉 = 〈wl ⊗ e, a⊗ u〉 = 〈a, wl〉〈e, u〉 = 〈a, wl〉. (20)
We can now finally show that a useful expression can be obtained for the values of any bilinear
mapping
b:∧qeV
∗ × ∧q
∗ → ∧mo V
∗. (21)
We associate with b the linear mappings
b:∧qeV
∗ → ∧q
e V ⊗ ∧
∗: a 7→ iq′(b(a, ·)) (22)
∗ → ∧qeV ⊗ ∧
∗: a′ 7→ iq(b(·, a
′)). (23)
Proposition 2. If b as in (21) is a bilinear mapping and b, b are the associated linear mappings (22)
and (23), then
b(a, a′) = a′ ∧ Weq′(b(a)) = a ∧ Weq(b(a
′)). (24)
for each (a, a′) ∈ ∧qeV
∗ × ∧q
Proof: Applying Lemma 3 to l = b(a, ·) we get
b(a, a′) = b(a, ·)(a′) = a′ ∧ Weq′(iq′(b(a, ·))) = a
′ ∧ Weq′(b(a)). (25)
The other equality is obtained in the same way by applying Lemma 3 to l = b(·, a′).
4. Integration of differential forms. Chains and currents.
Let M be an affine space modelled on a vector space V . A differential q-form on M is a differentiable
function A:M ××qV × O(V ) → R depending on a point, q vectors and an orientation. It is q-linear
and totally antisymmetric in its vector arguments. A differential form A is said to be even or odd if
for each point in x ∈ M it defines a even or odd multicovector, respectively. We note that a zero-form
on M is a differentiable function f :M ×O(V ) → R. The vector space of even differential q-forms will
be denoted by Φqe(M) and space of odd differential q-forms will be denoted by Φ
o(M). The symbol
Φqp(M) will be used to denote either of the two spaces when the distinction is of no importance.
For the definition of the exterior product of two differential forms and for that of exterior differential
of a differential form, we refer the reader to [5]. Here we just recall that if both forms A and A′ are
even or both are odd, the product A ∧ A′ is even. In other cases the product is odd. The parity of
the differential dA of a q-form A is the same as the parity of the original form A. A q-form A can
be interpreted as a q-covector field Ã:M → ∧qpV
∗. The exterior product and the exterior differential
are extended to this alternative interpretation of forms. The left and right interior multiplications of
even and odd multivector fields with even and odd multicovector fields are defined point by point in
an obvious manner.
A cell of dimension q or a q - cell in M is a pair (χ, o), where χ is a differentiable mapping χ:Rq → M
and o is an orientation of V . For q = 0, R0 is the vector space {0} with a single element 0. Hence a
zero-cell in M is a pair of a point x ∈ M and an orientation of V . The integral of a q - form A on a
cell (χ, o) is the Riemann integral
(χ,o)
· · ·
A (χ(s1, . . . , sq),D1χ(s1, . . . , sq), . . . ,Dqχ(s1, . . . , sq), o) ds1 · · · dsq. (26)
The integral of a 0 - form f on a zero-cell (x, o) is the value
(x,o)
f = f(x, o). For each q we introduce
the space K(Xq(M)) of formal linear combinations of q - cells. The formal linear combinations turn
into real linear combinations if cells are identified with elements of K(Xq(M)). Integration of forms is
extended to linear combinations by linearity. Subspaces N pq (M) ⊂ K(Xq(M)) are defined as the sets
q (M) =
C ∈ K(Xq(M));
A = 0 for each A ∈ Φqp(M)
. (27)
Elements of the quotient spaces Ξpq(M) = K(Xq(M))
N pq (M) are called even chains or odd chains of
dimension q. We extend the sequence of even and odd chains to negative dimension q by defining the
spaces Ξpq(M) = {0} for each q < 0. A chain is said to be simple if it has a single cell as one of its
representatives. Integrals of q - forms on q - chains are well defined. The integral of a q - form A on the
class C of C ∈ K(Xq(M)) is the integral of A on C.
The boundary operator ∂ assigns to a chain C ∈ Ξ pq (M) its boundary ∂C ∈ Ξ
q−1(M). The boundary
of a simple chain represented by a q - cell (χ, o) is the chain represented by the combination
i=1(−1)
i−1((χ(i,1), o) − (χ(i,0), o)), (28)
where the (q−1) - cells (χ(i,1), o) and (χ(i,0), o) are defined by
(i,∗):Rq−1 → M : (s1, . . . , ŝi, . . . , sq) 7→ χ(s1, . . . , si−1, ∗, si+1 . . . , sq) (29)
with ∗ replaced by 1 and 0, respectively. The cells introduced in (29) represent the faces of the simple
chain. The construction of the boundary is extended to generic chains by linearity. The boundary of
a boundary is the zero chain. It is known that Stokes theorem holds for chains and forms of the same
parity.
An even or odd de Rham current of dimension q on a manifold M is a linear function c: Φqp(M) → R.
We will use the symbol
A to denote the value c(A). The spaces of forms are given certain topologies
and the continuity of the function is required. Chains will be treated as currents. They form a dense
subspace in the space of currents. We will consider only very simple examples of currents other than
chains. Topological considerations are of little importance for these examples. The boundary of a
current is defined by assuming that Stokes theorem holds for currents. Thus if c is a current of
dimension q, then the boundary of c is the mapping
∂c: Φq−1p (M) → R:A 7→
dA. (30)
In addition to chains the odd de Rham current most frequently used is the Dirac current wδ(x) of
dimension m derived from an odd m - vector w and a point x ∈ M . If A is an odd m - form, then
wδ(x)
A = 〈Ã(x), w〉. (31)
B. Electrodynamics
1.The space of fields.
In this section we will construct a suitable space of fields for electrodynamics which has the same
role of the space of motions in mechanics (see [3]). The space of fields is not a differentiable manifolds.
Nevertheless, we will introduce in this space enough structure to permit the construction of tangent
and cotangent vectors which are the objects needed in in the formulation of the variational principle.
A class of admissible functions defined on the space of fields has also to be specified. This class needs
to be large enough to include the action, which for electrodynamics is generated (in a sense that will be
specified) from a quadratic Lagrangian density, due to the linearity of the theory. Thus, only functions
generated from quadratic mappings will be used in our variational formulation of electrodynamics. A
larger class of admissible functions would be needed to deal with more general, non linear field theories,
and the price of increased technical difficulties should be paid.
Let M be the affine Minkowski space-time of special relativity with the 4-dimensional model space
V and the non degenerate metric tensor g : V → V ∗ of signature (1, 3). The space of odd 4-currents
with compact supports in M will be denoted by CM . Differential forms will always be presented as
covector fields.
We consider the set X(Φ1e(M);CM) of pairs (A, c), where c is an odd current of dimension 4 in M
with a compact support Sup(c) and A is a local even 1 - form
A:U → ∧1eV
∗ (32)
defined in an open set U ⊂ M containing the support of c. The 1 - form A will represent the electro-
magnetic potential.
A mapping
κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗ (33)
is said to be quadratic if for each x ∈ M there exists a symmetric bilinear mapping
∗ × ∧2eV
∗ × ∧2eV
∗) → ∧4oV ∗ (34)
such that the mappings κx = κ(x, ·, ·) and δ
2κx satisfy the equality
κx(a, f) =
κx((a, f)(a, f)), (35)
for each (a, f) ∈ ∧1eV
∗ × ∧2eV
∗. We are using the standard definition of quadratic mappings in terms
of polarizations. The polarization of the mapping κx is the mapping δ
2κx and the equation (35) is the
standard relation between a quadratic mapping defined on a vector space and its polarization. We will
use the set of quadratic mappings (33) to introduce an equivalence relation in the set X(Φ1e(M);CM).
In the next section they will be also used to define the admissible functions defined on the space of
fields.
Pairs (A, c) and (A′, c′) are equivalent if
κ ◦ (x,A′, dA′) =
κ ◦ (x,A, dA) (36)
for each quadratic mapping κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗. The symbol x is used to indicate the
identity mapping of M and also a point of M .
If µ is an arbitrary odd 4-form on M and κ is set to be the mapping
κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (x, a, f) 7→ µ(x), (37)
then the equivalence condition (36) reduces to
µ (38)
and implies that c′ = c.
Equivalence classes of elements of X(Φ1e(M);CM) will be called fields. Our fields are similar to
those used by Freed in [14]. The space of fields will be denoted by Q(Φ1e(M);CM) or simply Q. The
equivalence class of (A, c) will be denoted by q(A, c) or simply q. There is a natural projection
ε:Q(Φ1e(M);CM) → CM : q(A, c) 7→ c (39)
from the space of fields to the space CM of currents in M . It is not difficult to check that each fibre
ε−1(c) of the projection ε is a vector space which will be denoted by the symbol Q(Φ1e(M); c) or Qc.
Indeed, the sum of two pairs (A, c) and (A′, c) with A:U → ∧1eV
∗ and A′:U ′ → ∧1eV
∗ is the pair
(A + A′, c) defined in U ∩ U ′ ⊃ Sup(c). Descending to the quotient with respect to the equivalence
relation defined by (36) easily gives the sum in Qc. Therefore, the space of fields is the disjoint union
of vector spaces Q =
2.Functions, vertical tangent vectors and covectors in the space of fields.
With each quadratic mapping κ:M × ∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗ we associate the function
k:Q(Φ1e(M);CM) → R: q(A, c) 7→
κ ◦ (x,A, dA). (40)
The above mentioned admissible functions defined on the space of fields are those constructed in this
way. They will be called differentiable. The space of such functions will be denoted by K(Φ1e(M);CM).
We note that from the definition of the space of fields Q = Q(Φ1e(M);CM) (as a quotient space with
respect to the equivalence condition (36)) follows easily that the differentiable functions separate points
of Q, i.e. if k(q′) = k(q) for each k ∈ K(Φ1e(M);CM), then q
′ = q. Indeed if k(q(A′, c′)) = k(q(A, c)) for
each k ∈ K(Φ1e(M);CM), then equation (36) holds for each quadratic mapping κ:M×∧
∗×∧2eV
∗. It follows that c′ = c and q(A′, c) = q(A, c).
The tangent space to a vector space Qc (i.e. to a fibre of ε) coincides with Qc.
The vertical tangent bundle to the vector bundle Q(Φ1e(M);CM) is the space
VQ = Q ×
(ε,ε)
Q = {(q, δq) ∈ Q×Q; ε(q) = ε(δq)} (41)
where q = q(A, c) and δq = q(δA, c) and A:U → ∧1eV
∗, δA:U → ∧1eV
∗. The tangent projection is the
canonical projection
τQ:VQ → Q: (q, δq) 7→ q. (42)
There is no obvious choice of the bundle dual to VQ.
We will use the fibre derivatives of functions k ∈ K(Φ1e(M);CM) as models of covectors. The
derivative Dk of a function k:Q → R is defined for each current c separately. It is evaluated on a pair
of vectors q = q(A, c) ∈ Qc and δq = q(δA, c) ∈ Qc. The result is the expression
Dk (q(A, c), q(δA, c)) =
k (q(A + sδA, c)) (0)
κ ◦ (x,A + sδA, F + sδF )
, (43)
where F = dA and δF = dδA. The symbol dk(q(A, c)) will be used to denote the covector characterized
by the pairing
〈dk (q (A, c)) , q(δA, c)〉 = Dk (q(A, c), q(δA, c)) (44)
with vectors δq = q(δA, c) ∈ Qc.
So we need to calculate the expression
κ ◦ (x,A + sδA, F + sδF )
(x) = Dκx(A(x), F (x), δA(x) ⊕ δF (x)), (45)
for each x ∈ U . We are using the following known definition. If V1, V2,W are vector spaces and
m:V1 × V2 → W is a smooth mapping, then its derivative is the mapping
Dm:V1 × V2 × (V1 ⊕ V2) → W : (a, f, δa⊕ δf) 7→
(m(a + sδa, f + sδf)) (0). (46)
We define for each x ∈ U the bilinear mappings
∗ × ∧1eV
∗ → ∧4oV
∗: (a, a′) 7→ δ2κx((a, 0), (a
, 0)), (47)
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ δ2κx((a, 0), (0, f)) (48)
∗ × ∧2eV
∗ → ∧4oV
∗: (f, f ′) 7→ δ2κx((0, f), (0, f
′)), (49)
obtaining the equality
κx(a, f) =
λx(a, a) + µx(a, f) +
νx(f, f), (50)
for each (a, f) ∈ ∧1eV
∗ × ∧2eV
∗. The equality (50) in terms of the bilinear mappings (47), (48), and
(49) will be useful to calculate the expression (45).
We have the following lemmas.
Lemma 4. If κx:∧
∗ × ∧2eV
∗ → ∧4oV
∗ is a quadratic mapping and λx, µx, νx are the mappings
defined above, then
Dκx(a, f, δa⊕ δf) =
Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) +
Dνx(f, f, δf ⊕ δf), (51)
for each (a, f, δa⊕ δf) ∈ ∧1eV
∗ × ∧2eV
∗ × ∧1eV
∗ ⊕ ∧2eV
Proof: We have the equality
λx ◦ ∆ ◦ pr1 + µx +
νx ◦ ∆
′ ◦ pr2, (52)
where
pr1:∧
∗ × ∧2eV
∗ → ∧1eV
∗: (a, f) 7→ a, (53)
∆:∧1eV
∗ → ∧1eV
∗ × ∧1eV
∗: a 7→ (a, a), (54)
pr2:∧
∗ × ∧2eV
∗ → ∧2eV
∗: (a, f) 7→ f, (55)
∆′:∧2eV
∗ → ∧2eV
∗ × ∧2eV
∗: a 7→ (a, a). (56)
The claim follows from the equality (52) by noting that
Dκx(a, f, δa⊕ δf) =
Dλx ◦ (∆,D∆) ◦ (pr1,Dpr1)(a, f, δa⊕ δf) + Dµx(a, f, δa⊕ δf)
Dνx ◦ (∆
,D∆′) ◦ (pr2,Dpr2)(a, f, δa⊕ δf)
Dλx ◦ (∆,D∆)(a, δa) + Dµx(a, f, δa⊕ δf)
Dνx ◦ (∆
′,D∆′)(f, δf)
Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) +
Dνx(f, f, δf ⊕ δf). (57)
Lemma 5. If V1, V2,W are vector spaces and
b:V1 × V2 → W (58)
is a bilinear mapping, then
Db(a, f, δa⊕ δf) = b(a, δf) + b(δa, f), (59)
for each (a, f, δa⊕ δf) ∈ V1 × V2 × (V1 ⊕ V2).
Proof: From the definition (46) of the derivative it follows that
Db(a, f, δa⊕ δf) =
(b(a, f + sδf) + sb(δa, f + sδf)) (0)
b(a, f) + sb(a, δf) + sb(δa, f) + s2b(δa, δf)
= b(a, δf) + b(δa, f). (60)
In view of Lemmas 4 and 5 the integrand in right hand side of the equality (43) reduces to
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) =
Dλx(A(x), A(x), δA(x) ⊕ δA(x))
+ Dµx(A(x), F (x), δA(x) ⊕ δF (x))
Dνx(F (x), F (x), δF (x) ⊕ δF (x))
(λx(A(x), δA(x)) + λx(δA(x), A(x)))
+ µx(A(x), δF (x)) + µx(δA(x), F (x))
(νx(F (x), δF (x)) + νx(δF (x), F (x))) , (61)
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) = λx(A(x), δA(x)) + µx(A(x), δF (x))
+ µx(δA(x), F (x)) + νx(F (x), δF (x)), (62)
since λx and νx are symmetric. By using the Proposition 2 contained in Section 6, this expression
reduces to
Dκx(A(x), F (x), δA(x) ⊕ δF (x)) =δA(x) ∧ We1(λx(A(x)) + δF (x) ∧ We2(µx(A(x))
+ δA(x) ∧ We1(µx(F (x)) + δF (x) ∧ We2(νx(F (x))
=δA(x) ∧
We1(λx(A(x)) + µx(F (x)))
+ δF (x) ∧ (We2(µx(A(x)) + νx(F (x)))) . (63)
where the linear mappings λx and λx are obtained from the bilinear mapping λx as prescribed in
Section 6, i.e.,
∗ → ∧1eV ⊗ ∧
∗: a 7→ i1(λx(a, ·)), (64)
∗ → ∧1eV ⊗ ∧
∗: a 7→ i1(λx(·, a)), (65)
for each x ∈ U . The mappings µx, µx are obtained from µx, and νx, νx are obtained from νx in the
same way.
We define from λx, λx the mappings
λ:M × ∧1eV
∗ → ∧1eV ⊗ ∧
∗: (x, a) 7→ λx(a), (66)
λ:M × ∧1eV
∗ → ∧1eV ⊗ ∧
∗: (x, a) 7→ λx(a). (67)
The mappings µ, µ, ν, ν are defined respectively from µx, µx, νx, νx in the same way.
By using the equalities (47) and (63) which hold for each x ∈ U we obtain the equality
κ ◦ (x,A + sδA, F + sδF )
We1 ◦
λ ◦ (x,A) + µ ◦ (x, F )
+ (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (68)
We have also used the graded commutativity of the exterior product.
Hence,
Dk (q(A, c), q(δA, c)) =
κ ◦ (x,A + sδA, F + sδF )
We1 ◦
λ ◦ (x,A) + µ ◦ (x, F )
(We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (69)
Since F = dA and δF = dδA, this equality can be converted to the form
Dk (q(A, c), q(δA, c)) = −
We1 ◦
λ ◦ (x,A) + µ ◦ (x, dA)
+ d (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, dA)))) ∧ δA
d ((We2 ◦ (µ ◦ (x,A) + ν ◦ dA)) ∧ δA) . (70)
We note that the first integral contains the exterior product of δA and an odd 3-form, while the second
integral contains the exterior differential of the product of δA and an odd 2-form.
The obtained expression suggests the representation of covectors as equivalence classes of elements
of the sets X(Φ2o(M) × Φ
o(M);CM) introduced below.
An element of X(Φ2o(M)×Φ
o(M);CM) is a triple (G, J, c) of a local odd differential 2-form G:U →
∗, a local odd differential 3-form J :U → ∧3oV
∗ and a current c with support contained in U . The
odd 2-form G will be interpreted as the electromagnetic induction, while the odd 3-form J will be
interpreted as the current. A covector p is an equivalence class p(G, J, c) of (G, J, c) ∈ X(Φ2o(M) ×
Φ3o(M);CM). The equivalence relations in X(Φ
o(M) × Φ
o(M);CM) are based on the expression
J ∧ δA−
d (G ∧ δA)
. (71)
Elements (G, J, c) and (G′, J ′, c′) are equivalent if c = c′ and
J ′ ∧ δA−
d (G′ ∧ δA)
J ∧ δA−
d (G ∧ δA)
, (72)
for each (δA, c) ∈ Qc.
This equivalence relation is related to the equivalence relation (36). They can be used in the
construction of dual objects. Indeed a (vertical) tangent vector might also (more generally) be defined
as an equivalence class of pairs (δA, c) with respect to an equivalence relation similar to (36). A
convenient representation of such relation is a relation similar to (72), but with (G, J, c) fixed, defining
the equivalence between two pairs (δA, c) and (δA′, c). Such a construction is not needed in our case
since each Qc is a vector space, so that its tangent space is canonically identified with Qc itself.
The vector space Πc of covectors associated with the current c will be used as the dual space of the
space Qc thought of as the space of vertical tangent vectors with the pairing
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
, (73)
where δq = q(δA, c) ∈ Qc and p = p(G, J, c) ∈ Πc. The space of all covectors is the union
Πc. (74)
There is a natural projection
ε′: Π → CM : p(G, J, c) 7→ c. (75)
from the space of fields to the space CM of currents in M . The phase space is the space
Ph = Q ×
(ε,ε′)
Π = {(q, p) ∈ Q× Π; ε(q) = ε′(a)}. (76)
The symbol Phc will denote the set Qc × Πc ⊂ Ph.
3.A virtual action principle for electrodynamics.
In this section a variational principle for electrodynamics similar to the virtual action principle of
analytical mechanics (see [3]) will be formulated. The construction is an example of similar construc-
tions needed for the variational formulation of a general field theory. The linearity of the theory and
the choice of formulating it on the affine Minkowski space makes our presentation simpler though
containing the core of the general framework (which can be found in [1]).
The action is the differentiable function
W :Q(Φ1e(M);CM) → R: q(A, c) 7→
L ◦ (A, dA) (77)
derived from the quadratic Lagrangian density
L:∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ −
〈f,∧2eg
−1(f)〉
|g|. (78)
We are denoting by
|g| the odd 4-covector defined by the Minkowski metric (see [5]). Later we
will use the same symbol to denote the constant 4-covector field constructed from it.
The 1-form A is called the potential, the 2-form F = dA is called the electromagnetic field.
We note that the Lagrangian is a quadratic mapping which depends only on its second argument f
and thus λ = µ = 0 and ν does not depend on x in the formula (50).
A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if the equality
〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉
= 0 (79)
holds for each virtual displacement δq = q(δA, c) ∈ Qc. For each current c the dynamics associated
with the current c is the set Dc ⊂ Phc of phases which satisfy the virtual action principle. The
dynamics is the subset
Dc (80)
of the phase space Ph.
A phase space trajectory is a triple of local differential forms
(A,G, J):U → ∧1eV
∗ × ∧2oV
∗ × ∧3oV
. (81)
The dynamics of a system can also be represented as a set DDDD of phase space trajectories (A,G, J) such
that for each current c with support included in U the phase ph = (q(A, c), p(G, J, c)) is in Dc.
The equation (79) is too abstract to be used directly. A more concrete expression is given in the
following proposition.
Proposition 3. A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if and only if
the equality
−1 ◦ dA
∧ δA− d
−1 ◦ dA
J ∧ δA−
d (G ∧ δA)
is satisfied for each virtual displacement q(δA, c).
Proof: By applying the formula (70) to the action W we obtain that its variation is
〈dW (q), δq〉 =
(−d(We2 ◦ ν ◦ dA) ∧ δA + d ((We2 ◦ ν ◦ dA) ∧ δA)) , (83)
since λ = µ = 0. Moreover
νx(f, f) = −
〈f,∧2eg
−1(f)〉
|g|, (84)
from which it follows after some calculation that
ν(x, f) = −
∧2e g
−1(f) ⊗
|g|, (85)
for each f ∈ ∧2eV
∗. Hence,
We2 ◦ ν ◦ (x, dA) = −
−1 ◦ dA
and the variation of the action reduces to
〈dW (q), δq〉 =
−1 ◦ dA
∧ δA− d
−1 ◦ dA
. (87)
Recalling that, on the other hand
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
, (88)
the claim follows.
A phase space trajectory belongs to the dynamics DDDD, if and only if it satisfies the virtual action
principle for each current c with support included in its domain of definition. There is a characterization
of the dynamics of phase space trajectories in terms of differential equations. This is shown in the
following propositions.
Theorem 4. A phase space trajectory (A,G, J) belongs to the dynamics DDDD if and only if it satisfies
the Euler-Lagrange equation
−1 ◦ dA
J (89)
and the constitutive relation
−1 ◦ dA
|g|. (90)
Proof: If a phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the constitutive
relation, then by substituting the expressions (89) and (90) of J and G in terms of the electromagnetic
field F in the virtual action principle (82) it follows that (A,G, J) belongs to the dynamics DDDD.
The inverse implication will be proved in the next section.
The constitutive relation (90) produced by our variational principle corresponds to the momentum-
velocity relation of analytical mechanics.
Proposition 5. A phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the
constitutive relation if and only if it satisfies the Maxwell’s equations
J (91)
and the constitutive relation
−1 ◦ F
|g|, (92)
with F = dA.
Proof: The constitutive relation (91) is satisfied if and only if the equation (90) is satisfied for
F = dA.
If a phase space trajectory (A,G, J) satisfies the Maxwell’s equations and the constitutive relation,
then by substituting the expression (92) of the electromagnetic induction in the equation (91) we see
that the Euler-Lagrange equation is satisfied, since F = dA.
Conversely if the Euler-Lagrange equation and the constitutive relation are satisfied, then, again by
substitution, we obtain that the Maxwell’s equations (91) are satisfied.
We remark that since the virtual action principle (77) is more complete than the Hamilton principle,
our formulation leads to the Maxwell’s equations (91) with external sources. The proposed variational
principle also permits the derivation of the constitutive relations (92) which are usually postulated
separately since the variations normally considered are not general enough to derive them from the
variational principle.
4.The Dynamics in a compact domain.
Let the current c consist in integrating an odd 4-form on a compact domain K ⊂ M with smooth
boundary ∂K. Field configurations, tangent vectors and covectors are equivalence classes of equivalence
relations based on the expressions ∫
κ ◦ (x,A, dA) (93)
and ∫
J ∧ δA−
d (G ∧ δA)
J ∧ δA−
G ∧ δA. (94)
It follows that a field q = q(A,K) is represented by the restriction
A|K:K → ∧1eV
∗ (95)
of the potential A to the the domain K. A tangent vector δq = q(δA,K) is represented by the
restriction
(δA)|K:K → ∧1eV
∗ (96)
of the variation δA to the domain K. A covector p = p(G, J,K) is represented by the pair of the
restriction
G|∂K: ∂K → ∧2oV
∗ (97)
of the electromagnetic induction G to the boundary ∂K of the domain K and the restriction
K → ∧3oV
∗ (98)
of the current J to the interior
K of the domain K.
A phase is a pair (q, p) of a field q = q(A,K) and a covector p = p(G, J,K).
The action is
W :QK → R: q(A,K) 7→
L ◦ (A, dA). (99)
The virtual action principle is the equality
〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉
= 0 (100)
and the dynamics in the domain K is the set DK ⊂ Ph of phases satisfying the virtual action principle.
Proposition 6. A phase ph = (q(A,K), p(G, J,K)), defined in a compact domain K, belongs to the
dynamics DK if and only if the Euler-Lagrange equation
−1 ◦ dA
K (101)
and the constitutive relation
G|∂K =
−1 ◦ dA
|∂K (102)
are satisfied.
Proof: If q = q(A,K), p = p(G, J,K) and δq = q(δA,K), then
〈dW (q), δq〉 =
−1 ◦ dA
−1 ◦ dA
−1 ◦ dA
−1 ◦ dA
∧ δA. (103)
On the other hand
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
G ∧ δA. (104)
Thus the virtual action principle assumes the form
−1 ◦ dA
∧ δA−
−1 ◦ dA
J ∧ δA−
G ∧ δA. (105)
By using variations with (δA)|∂K = 0 we derive the Euler-Lagrange equation
−1 ◦ dA
K. (106)
Assuming that this equation is satisfied and using arbitrary variations, the constitutive relation
G|∂K =
−1 ◦ dA
|∂K (107)
follows.
The following Proposition completes the proof of the Theorem 4.
Proposition 7. If a phase space trajectory
(A,G, J):U → ∧1eV
∗ × ∧2oV
∗ × ∧3oV
. (108)
belongs to the dynamics DDDD, then the Euler-Lagrange equation
−1 ◦ dA
J (109)
and the constitutive relation
−1 ◦ dA
|g| (110)
are satisfied in U .
Proof: If (A,G, J) is a phase space trajectory, defined in the open set U ⊂ M , which belongs to the
dynamics DDDD, then the equation
−1 ◦ dA
K (111)
and the boundary relation
G|∂K =
−1 ◦ dA
|∂K (112)
are satisfied for each compact domain K ⊂ U . It follows that equations (109) and (110) are satisfied
in every x ∈ U .
5.The Lagrangian formulation of electrodynamics.
The Lagrangian formulation of dynamics is the infinitesimal limit of the formulation in a compact
domain with the domain shrinking to a point. A formal method which greatly simplifies the passage
to the infinitesimal limit is to replace the compact domain — which is used exclusively as domain
of integration — with the current c = δ(x)w, where δ(x) is the Dirac delta function in x ∈ M and
w ∈ ∧4oV is an odd 4-vector, with w 6= 0. The construction of infinitesimal fields, tangent vectors and
covectors is based on the expressions
δ(x)w
κ ◦ (x,A, dA) = 〈κ (x,A(x), dA(x)) , w〉 (113)
δ(x)w
J ∧ δA−
d (G ∧ δA)
J(x) ∧ δA(x) −
d(G ∧ δA)(x), w
. (114)
Since w 6= 0 and ∧4oV is one-dimensional, it follows from the first expression that a field q = q(A, c) is
represented by the pair
(A(x), F (x)) ∈ ∧1eV
∗ × ∧2eV
∗ (115)
of an even 1-covector A(x) and an even 2-covector F (x).
The second expression reduces to
dG(x) −
∧ δA(x) + G(x) ∧ δF (x), w
, (116)
since dδA(x) = δF (x).
It follows that a tangent vector δq = q(δA, c) is represented by the pair
(δA(x), δF (x)) ∈ ∧1eV
∗ × ∧2eV
∗ (117)
and a covector p = p(G, J, c) is represented by the pair
G(x), dG(x) −
∈ ∧2oV
∗ × ∧3oV
∗. (118)
The pairing
〈〈〈〈〈p, δq〉〉〉〉〉
J ∧ δA−
d (G ∧ δA)
(119)
assumes the form
〈〈〈〈〈p, δq〉〉〉〉〉
dG(x) −
∧ δA(x) + G(x) ∧ δF (x), w
. (120)
We have constructed the space of infinitesimal fields Qδ = ∧
∗×∧2eV
∗ and the space of infinitesimal
covectors Πδ = ∧
∗ × ∧3oV
∗. Hence, the infinitesimal phase space is
Phδ = Qδ × Πδ = ∧
∗ × ∧2eV
∗ × ∧2oV
∗ × ∧3oV
∗. (121)
The infinitesimal action is
W (q(A, δ(x)w)) = 〈L(A(x), F (x)), w〉 . (122)
The infinitesimal dynamics is the set
(q, δq) ∈ Phδ; ∀δq∈Qδ 〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉
, (123)
It is easy to verify that the infinitesimal dynamics Dδ admits also the following more explicit expression
(a, f, g, h) ∈ Phδ; ∀
(δa,δf)∈∧1eV
∗×∧2eV ∗
DL(a, f, δa, δf) = −
(h ∧ δa + g ∧ δf)
. (124)
The infinitesimal dynamics Dδ is characterized by the following Proposition.
Proposition 8. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to
the infinitesimal dynamics Dδ if and only if the equations
G(x) =
−1(F (x))
|g| (125)
dG(x) =
J(x) (126)
are satisfied.
Proof: If q = q(A, δ(x)w)), p = p(G, J, δ(x)w), and δq = q(δA, δ(x)w), with w 6= 0, then the variation
of the action (87), which can also be expressed in the form
〈dW (q), δq〉 = −
−1 ◦ F
∧ δF, (127)
reduces to
〈dW (q), δq〉 = −
−1(F (x))
∧ δF (x), w
. (128)
Thus the virtual action principle
〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉
(129)
has the explicit form
dG(x) −
∧ δA(x) +
G(x) −
−1(F (x))
∧ δF (x), w
= 0. (130)
Therefore if ph satisfies the equations (125) and (126), then (130) is satisfied and ph ∈ Dδ.
Conversely, if ph ∈ Dδ, then it satisfies the virtual action principle, which implies
dG(x) −
∧ δA(x) +
G(x) −
−1(F (x))
∧ δF (x) = 0, (131)
since w 6= 0 and ∧4oV is one-dimensional. The equations (125) and (126) follow, since δA(x) and δF (x)
are independent.
In a preceding section we showed that the dynamics of phase space trajectories can also be charac-
terized by the Maxwell’s equations and the constitutive relation. This fact can now be proved directly.
Indeed, if a phase space trajectory (A,G, J) belongs to the dynamics DDDD and x is a point in the
domain of definition of the trajectory, then the virtual action principle is satisfied for every infinitesimal
current δ(x)w. Thus the Maxwell’s equations and the constitutive relation are satisfied in x, thanks
to the previous proposition. It follows that they are satisfied in the whole domain of definition of the
trajectory.
To prove the inverse implication, we observe that the virtual action principle (82) can also be
expressed in the form
−1 ◦ F
|g| ∧ δF =
J ∧ δA−
d (G ∧ δA)
(132)
−1 ◦ F
∧ δF +
= 0, (133)
for each virtual displacement q(δA, c). Thus if a phase space trajectory (A,G, J) satisfies the Maxwell’s
equations, then it satisfies the virtual action principle for every current c with support contained in
the domain of definition of the trajectory and hence it belongs to DDDD.
6.The Hamiltonian formulation of electrodynamics.
We associate with the Lagrangian density
L:∧1eV
∗ × ∧2eV
∗ → ∧4oV
∗: (a, f) 7→ −
f,∧2eg
−1(f)
|g|, (134)
the energy density
E:∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗ → ∧4oV
∗ (135)
defined by
E(a, g, f) = −
g ∧ f − L(a, f)
g ∧ f +
f,∧2eg
−1(f)
g ∧ f +
−1(f)
2g − ∧2eg
−1(f)
∧ f (136)
and treat this mapping as a family
E(a, g, ·):∧2eV
∗ → ∧4oV
∗ (137)
of mappings on the fibres of the projection
prP :∧
∗ × ∧2oV
∗ × ∧2eV
∗ → ∧1eV
∗ × ∧2oV
∗ (138)
onto the field-momentum space P = ∧1eV
∗ × ∧2oV
∗. The set
Cr(E, prP ) =
(a, g, λ) ∈ ∧1eV
∗ × ∧2oV
∗ × ∧2eV
δλ∈∧2
DE(a, g, λ, 0, 0, δλ) = 0
(139)
is the critical set of the family. The equality
DE(a, g, λ, δa, δg, δλ) = −
(δg ∧ λ + g ∧ δλ) − DL(a, λ, δa, δλ)
(δg ∧ λ + g ∧ δλ) +
−1(λ)
δg ∧ λ +
−1(λ)
(140)
implies
DE(a, g, λ, 0, 0, δλ) = −
−1(λ)
∧ δλ. (141)
Thus we obtain the expression
Cr(E, prP ) =
(a, g, λ) ∈ ∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗; g = ∧2eg
−1(λ)
. (142)
The critical set is the graph of the Legendre mapping
Λ:∧1eV
∗ × ∧2eV
∗ → ∧2oV
∗: (a, λ) 7→ ∧2eg
−1(λ)
|g|. (143)
For each a ∈ ∧1eV
∗, the mapping Λ(a, ·) is invertible. Its inverse is the mapping
Λ(a, ·)−1:∧2oV
∗ → ∧2eV
∗: g 7→ ∧2eg
|g−1| g
, (144)
where
|g−1| denotes the odd 4-vector characterized by 〈
|g−1|〉 = 1. It follows that the critical
set is the image of the section
σ:∧1eV
∗ × ∧2oV
∗ → ∧1eV
∗ × ∧2oV
∗ × ∧2eV
∗: (a, g) 7→
a, g,∧2eg
|g−1| g
(145)
of the projection prP . The Hamiltonian density is the mapping
H = E ◦ σ:∧1eV
∗ × ∧2oV
∗ → ∧4oV
∗, (146)
defined by the formula
H(a, g) = −
|g−1| g
, (147)
for each (a, g) ∈ ∧1eV
∗×∧2oV
∗. The passage from the Lagrangian density L to the Hamiltonian density
H is the Legendre transformation of electrodynamics.
We show that the energy density can be used to generate the infinitesimal dynamics Dδ. We consider
the set
(a, f, g, r) ∈ Phδ; ∃
V ∗ ∀(δa,δg,δλ)∈∧1
V ∗×∧2
V ∗×∧2
DE(a, g, λ, δa, δg, δλ) =
(r ∧ δa− f ∧ δg)
. (148)
This set is obtained by projecting the set
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
(δa,δg,δλ)∈∧1eV
∗×∧2oV ∗×∧2eV ∗
DE(a, g, λ, δa, δg, δλ) =
(r ∧ δa− f ∧ δg)
(149)
onto the phase space Phδ = ∧
∗ × ∧2eV
∗ × ∧2oV
∗ × ∧3oV
It follows from (140) that λ = f and that the set D̃E reduces to
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
∗; λ = f,
(δa,δg,δf)∈∧1eV
∗×∧2oV ∗×∧2eV ∗
DL(a, f, δa, δf) =
(r ∧ δa + g ∧ δf)
. (150)
It projects onto the infinitesimal dynamics
(a, f, g, r) ∈ Phδ; ∀
(δa,δf)∈∧1
V ∗×∧2
DL(a, f, δa, δf) = −
(r ∧ δa + g ∧ δf)
. (151)
Hence, DE = Dδ.
It is clear from the definition of the set D̃E that this set is included in the set
(a, f, g, r, λ) ∈ Phδ × ∧
∗; (a, g, λ) ∈ Cr(E, prP )
. (152)
The use of the mapping σ results in
D̃E =
(a, f, g, r, λ) ∈ Phδ × ∧
∗; λ = f,
(δa,δg)∈∧1
V ∗×∧2
DH(a, g, δa, δg) =
(r ∧ δa− f ∧ δg)
. (153)
The Hamiltonian description of the dynamics
(a, f, g, r) ∈ Phδ; ∀
(δa,δg)∈∧1
V ∗×∧2
DH(a, g, δa, δg) =
(r ∧ δa− f ∧ δg)
(a, f, g, r) ∈ Phδ; ∧
|g−1| g
= f , r = 0
. (154)
is obtained by projecting onto the phase space Phδ.
Proposition 9. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to
the infinitesimal dynamics Dδ if and only if the equations
F (x) = ∧2eg ◦
|g−1| G(x)
(155)
dG(x) =
J(x) (156)
are satisfied.
Proof: We recall that an infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, is
represented by (
A(x), F (x), dG(x) −
J(x), G(x)
(157)
and use the Hamiltonian description (154) of infinitesimal dynamics Dδ. The claim easily follows.
The resulting equations for the phase space trajectories (A,G, J)
F = ∧2eg ◦
|g−1| G
(158)
J (159)
are called Hamilton’s equations. The equations (159) are again Maxwell’s equations and the equation
(158) is the inverse of constitutive relation (92).
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