Balance Systems and the Variational Bicomplex
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the su...
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irk-123456789-1471852019-02-14T01:26:30Z Balance Systems and the Variational Bicomplex Preston, S. In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental ''pure non-Lagrangian'' balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the ''pure non-Lagrangian'' systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947-948] and, later, asserted as the canonical hyperbolic form of balance systems in [Müller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998]. 2011 Article Balance Systems and the Variational Bicomplex / S. Preston // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 49Q99; 35Q80 DOI:10.3842/SIGMA.2011.063 http://dspace.nbuv.gov.ua/handle/123456789/147185 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental ''pure non-Lagrangian'' balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the ''pure non-Lagrangian'' systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947-948] and, later, asserted as the canonical hyperbolic form of balance systems in [Müller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998]. |
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Preston, S. |
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Preston, S. Balance Systems and the Variational Bicomplex Symmetry, Integrability and Geometry: Methods and Applications |
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Preston, S. |
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Preston, S. |
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Balance Systems and the Variational Bicomplex |
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Balance Systems and the Variational Bicomplex |
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Balance Systems and the Variational Bicomplex |
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Balance Systems and the Variational Bicomplex |
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Balance Systems and the Variational Bicomplex |
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balance systems and the variational bicomplex |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/147185 |
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Balance Systems and the Variational Bicomplex / S. Preston // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT prestons balancesystemsandthevariationalbicomplex |
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2025-07-11T01:33:54Z |
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2025-07-11T01:33:54Z |
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1837312396667912192 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 063, 18 pages
Balance Systems and the Variational Bicomplex?
Serge PRESTON
Department of Mathematics and Statistics, Portland State University,
Portland, OR, 97207-0751, USA
E-mail: serge@pdx.edu
URL: http://www.mth.pdx.edu/~serge/
Received January 27, 2011, in final form June 30, 2011; Published online July 09, 2011
doi:10.3842/SIGMA.2011.063
Abstract. In this work we show that the systems of balance equations (balance systems) of
continuum thermodynamics occupy a natural place in the variational bicomplex formalism.
We apply the vertical homotopy decomposition to get a local splitting (in a convenient do-
main) of a general balance system as the sum of a Lagrangian part and a complemental “pure
non-Lagrangian” balance system. In the case when derivatives of the dynamical fields do not
enter the constitutive relations of the balance system, the “pure non-Lagrangian” systems
coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947–948]
and, later, asserted as the canonical hyperbolic form of balance systems in [Müller I., Rug-
geri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy,
Vol. 37, Springer-Verlag, New York, 1998].
Key words: variational bicomplex; balance equations
2010 Mathematics Subject Classification: 49Q99; 35Q80
1 Introduction
Systems of balance equations (balance systems) are a cornerstone of the mathematical models
of continuum thermodynamics. That is why the usage of variational methods for the study of
such systems was attempted in the corresponding domains of classical physics for a long time.
For a review of the present state of this research field, we refer to the monographs [19, 20] and
the references therein. These works use the conventional setting of variational calculus based on
the action principle.
Infinitesimal alternatives to the action principle were suggested by A. Poincaré and É. Cartan
in mechanics, by C. Caratheodory, H. Weyl and Th. Lepage in field theory and gradually evolved
into a mathematical theory of a rare beauty, see, for instance, [1, 4, 9, 12]. Thus, it seems natural
to investigate and to develop methods of infinitesimal variational calculus that can be used in
the study of general balance systems. In the papers [13, 14] it was proved that any balance
system allows a natural realization by an appropriately modified Cartan form (see Section 3).
Recently, I’ve observed that the balance systems present the largest class of systems of partial
differential equations that can be realized using the forms of Lepage type on the infinite jet
bundle J∞(π) of the configurational bundle π [15].
One of the most successful approaches in variational calculus is the method of variational
bicomplex, see [1, 12]. In this paper we show that general systems of balance equations occupy
a natural place in the framework of variational bicomplex.
In Section 2 we introduce the necessary notions and notation. In Section 3 the balance
systems of order k are defined. We recall the construction of the modified Poincaré–Cartan
?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special
Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html
mailto:serge@pdx.edu
http://www.mth.pdx.edu/~serge/
http://dx.doi.org/10.3842/SIGMA.2011.063
http://www.emis.de/journals/SIGMA/S4.html
2 S. Preston
forms for these systems, leading to the intrinsic formulation of the system of balance equations.
An (n, 1)-form KC carrying all the constituents of a balance system is introduced.
In Section 4 we show that any balance system is realized by the functional form corresponding
to the (n, 1)-form KC . Higher versions of balance systems are introduced here as well.
In Section 5 we apply the vertical homotopy decomposition to get a local splitting of a ge-
neral (n, 1)-form KC into the Lagrangian part (corresponding to a function we call “quasi-
Lagrangian”) and a complemental “pure non-Lagrangian” balance system (K-splitting). Some
properties of this decomposition are presented, including induced splitting of corresponding
functional forms (F -splitting). Finally, we obtain the condition on the form KC to have zero
Lagrangian part.
In Section 6 we determine the conditions under which the Lagrangian part of the functional
form for a balance systems of order zero (when the derivatives of dynamical fields do not enter
the constitutive relation) vanishes. It is proved that in such a case, a “pure non-Lagrangian”
system is of the type introduced by S. Godunov in 1961 [5] and later asserted by T. Rug-
geri and his coauthors, as the canonical (symmetrical hyperbolic) form of balance systems in
rational extended thermodynamics [11]. We calculate the “quasi-Lagrangian” for the balance
systems of hyperelasticity and the 2-dimensional ideal plasticity and show that its input into
the corresponding balance system reduces to the source terms in the balance laws.
2 Settings and notations
Throughout this paper π : Y → X is a (configurational) fibre bundle with a n-dimensional
connected paracompact smooth C∞-manifold X as the base and the total space Y , dim(Y ) =
n+m. The fiber of the bundle π is a m-dimensional connected smooth manifold U .
The base manifold X is endowed with a pseudo-Riemannian metric G. The volume form of
metric G will be denoted by η. Our considerations are mostly local and, as a result, we shall
not deal with the boundary of X.
We will be using fibred charts (W,xµ, yi) in a domain W ⊂ Y . Here (π(W ), xµ) is the chart
in X and yi are coordinates along the fibers. A local frame corresponding to the chart (W,xµ, yi)
will be denoted by (∂µ = ∂xµ , ∂i = ∂yi) (the shorter notation will be used in more cumbersome
calculations) and the corresponding coframe (dxµ, dyi).
We introduce the contracted forms ηµ = i∂xµη. Below we will be using following relations for
the forms ην without references (here and below λG = ln(
√
|G|))
dxν ∧ ηµ = δνµη, dηµ = λG,µη.
Section s : V → Y of the bundle π, defined in a domain V ⊂ X, corresponds to the collection
{yi(x)} of classical fields. Usually these fields are components of tensor fields or tensor densities
fields (sections of “natural bundles” [4]).
For a manifold M we will denote by τM : T (M) → M the tangent bundle of a manifold M
and by τ∗M : T ∗(M)→M its cotangent bundle. For a bundle Y → X, denote by πV : V (π)→ Y
the subbundle of tangent bundle T (Y ) consisting of the vertical vectors i.e. vectors ξ ∈ Ty(Y )
such that π∗yξ = 0.
Denote by Ωr(M) → M the bundle of exterior r-forms on a manifold M and by Ω∗(M) =
⊕∞r=0Ωr(M) the differential algebra of exterior forms on the manifold M .
Given a fiber bundle π : Y → X denote by Jk(π) the k-jet bundle of sections of the bun-
dle π [8, 16]. For a section s : U → Y , denote by jks : U → Jk(π) the k-jet of section s.
Denote by πkr : Jk(π) → Jr(π), k ≥ r ≥ 0 the natural projections between the jet bundles
of different order and by πk : Jk(π)→ X the projection to the base manifold X.
Balance Systems and the Variational Bicomplex 3
Projection mappings πk(k−1) : Jk(π)→ Jk−1(π) form the tower of k-jet bundles
· · · → Jk(π)→ Jk−1(π)→ · · · → Y → X. (2.1)
A bundle πk(k−1) : Jk(π) → Jk−1(π) has the structure of an affine bundles modeled on the
vector bundle SkT ∗(X)⊗→Jk−1(π) V (π)→ Jk−1(π).
Denote by J∞(π) the infinite jet bundle of bundle π – the inverse limit of the projective
sequence (2.1) [9, 12, 16]. Space J∞(π) is endowed with the structure of inverse limit of
differentiable manifolds with the natural sheaves of vector fields, differential forms etc., making
the projections π∞k : J∞(π) → Jk(π) the smooth surjections. See [8, 2, 16] for more about
structure and properties of k-jet bundles.
For a mutliindex I = {i1, . . . , in}, ik ∈ N denote by ∂I the differential operator ∂If =
∂i1
x1 · · · ∂inxn in C∞(X). To every fibred chart (W,xµ, yi) in Y there corresponds the fibred chart
(xµ, yi, ziµ, |I| =
∑
s is ≤ k) in the domain W k = π−1
k0 (W ) ⊂ Jk(π). Coordinates ziΛ in the
fibers of jet bundles are defined by the condition zµI (jkxs) = ∂Isµ(x).
For k = 1, . . . ,∞ the space Jk(π)→ X of k-jet bundle is endowed with the contact (Cartan)
distribution Cak defined by the basic contact forms
ωi = dyi − ziµdxµ, . . . , ωiI = dziI − ziI+1νdx
ν , |I| < k.
These forms generate the contact ideal Ck ⊂ Ω∗(Jk(π)) in the algebra of all exterior forms. A p-
form is called l-contact if it belongs to the l-th degree of this ideal (Ck)l ⊂ Ω∗(Jk(π)). 0-contact
exterior forms are also called horizontal (or πk-horizontal) forms (or, sometimes, semi-basic
forms).
Any exterior l-form ωl on the space Ω∞(π) has the decomposition ωl =
∑
s+r=l ω
(s,r) into its
s-horizontal, r-vertical components ω(s,r) ((s, r)-forms). Denote by Ω(s,r)(J∞(π)) the space of
all (s, r)-forms on J∞(π).
Bellow we will use the presentation of the exterior differential d as the sum of horizontal
and vertical differentials dH , dV : for a q-form ν ∈ Ω∗(Jk(π)) its differential dν lifted into the
Jk+1(π) is presented as the sum of horizontal and contact (vertical) terms: dν = dHν + dV ν,
see [9, 12]. Operators dH , dV are naturally defined in the space Ω∗(J∞)(π)).
We recall that these operators have the following homology properties
d2
h = d2
V = dV dH + dHdV = 0.
In particular, for a function f ∈ C∞(J∞(π)) (depending on the jet variables ziI up to some
degree, say |I| ≤ k),
df = (dµf)dxµ +
∑
|I|≥0
f,ziI
ωiI , (2.2)
where
dµf = ∂xµf +
∑
|I|≥0
ziI+1µ∂ziI
f (2.3)
is the total derivative of the function f(x, y, z) by xµ. The series in the formulas (2.2), (2.3)
contains finite number of terms: |I| ≤ k.
4 S. Preston
3 Balance systems, form KC and the Helmholtz coindition
Let π : Y → Xn be a configurational bundle and G a pseudo-Riemannian metric in X. Balance
systems of order k is the system of equations for the fields yi(x) of the form
dµF
µ
i + Fµi λG,µ = Πi, i = 1, . . . ,m. (3.1)
Here xµ, µ = 0, . . . , n are local coordinates in X – part of the fibred chart (W,xµ, yi) in Y , λG =
ln(
√
|G|). Densities F 0
i , fluxes FAi , A = 1, . . . , n and source/production terms Πi are smooth
functions on Jk(π), defined by the constitutive relations C, which determines the properties of
the physical system (solid, fluid, gas, etc.).
In the works [13, 14] we’ve associated with a balance systems (3.1) the n + (n + 1)-form
(modified Cartan form)
ΘC = Fµi ω
i ∧ ηµ + Πiω
i ∧ η + Fµi ω
i
µ ∧ η.
The Euler–Lagrange equations, corresponding to this form ΘC(
jk+1s
)∗
iξd̃ΘC = 0, ∀ ξ ∈ X
(
Jk(π)
)
,
coincide with the balance system (3.1). Here d̃ is the “con”-differential acting on a k+ (k+ 1)-
form αk + βk+1 as follows: d̃(αk + βk+1) = (−dα+ β) + dβ (see [13, 14]).
Now we restrict our consideration to the case of a balance system (3.1) of order one (i.e. Fµi ,
Πi ∈ C∞(J1(π))) and associate with such system the following (1-contact) (n, 1)-form on J2(π)
KC =
(
Fµi ω
i
µ + Πiω
i
)
∧ η. (3.2)
Next we obtain the following version of (local) Helmholtz condition (comp. [1]) for a balance
system of order one to be Euler–Lagrange system for some Lagrangian L ∈ C∞(J1(π)).
Proposition 1. For a constitutive relation C of order 1 the following properties are equivalent:
1) form KC is closed: dKC = 0,
2) C is locally Lagrangian constitutive relation: Fµi = ∂L
∂ziµ
, Πi = ∂L
∂yi
for the locally defined
function L ∈ C∞(J1(π)).
Proof. Let the form KC be closed. Calculate the differential of this form:
dKC =
[
(∂yjΠi)ω
j ∧ ωi + (∂yiF
µ
j − ∂zjµΠi)ω
i ∧ ωjµ + (∂
zjν
Fµi )ωjν ∧ ωiµ
]
∧ η
=
[
(∂yjΠi)dy
j ∧ dyi + (∂yiF
µ
j − ∂zjµΠi)dy
i ∧ dzjµ + (∂
zjν
Fµi )dzjν ∧ dziµ
]
∧ η.
The second line in last formula shows that if we consider Fµi and Πi as functions of vertical
variables yj , zjν (i.e. if we fix x) then the 1-form of these variables Fµi dz
i
µ + Πidy
i is closed.
Then, locally, by Poincaré lemma, Fµi dz
i
µ + Πidy
i = dV L for some function L(x, y, z). dV here
is the differential by the variables yj , zjν . It is easy to see now that KC = d(Lη).
Proof that the statement 1) follows from 2) is trivial. �
Remark 1. Since dKC = dVKC , the statement of Proposition 1, equivalent to the local equality
KC = dV (Lη), follows from the qualified exactness of the column of the augmented variational
bicomplex (see bellow), starting at Ωn,0, at the term Ωn,1.
Balance Systems and the Variational Bicomplex 5
4 Balance systems and the variational bicomplex
Proposition 1 for a balance system (3.1) to be Lagrangian points at the possibility to interpret
and study the balance systems using the variational bicomplex, see [1, 12]. In this section we
realize this possibility. In presenting the variational bicomplex we will follow [1].
The sheaf of k-contact s+k-forms on the infinite jet bundle J∞(π) is denoted by Ωs,k(J∞(π))
or, for shortness, Ωs,k.
We will be using the augmented variational bicomplex – the diagram where all the columns
and rows are complexes of sheaves of differential forms (right column is the complex of factor-
sheaves Fk = Im(I : Ωn,k(J∞(π))→ Ωn,s(J∞(π))).xdV xdV xδV
0 −−−−−→ Ω0,3 · · · Ωn,3
I−−−−−→ F3 −−−−−→ 0xdV xdV xδV
0 −−−−−→ Ω0,2 dH−−−−−→ Ω1,2 dH−−−−−→ · · · Ωn−1,2 dH−−−−−→ Ωn,2
I−−−−−→ F2 −−−−−→ 0
dV
x dV
x dV
x dV
x δV
x
0 −−−−−→ Ω0,1 dH−−−−−→ Ω1,1 dH−−−−−→ · · · Ωn−1,1 dH−−−−−→ Ωn,1
I−−−−−→ F1 −−−−−→ 0
dV
x dV
x dV
x dV
x
0 −−−−−→ R −−−−−→ Ω0,0 dH−−−−−→ Ω1,0 dH−−−−−→ · · · Ωn−1,0 dH−−−−−→ Ωn,0
(4.1)
Recall also the Euler–Lagrange mapping E :
E : Ωn,0 → F1, E([Lη]) = I(dV Lη),
where I : Ωn,s(J∞(π)) → Ωn,s(J∞(π)) is the “interior Euler (homotopy) operator” (see (4.2)
below). Mapping E is well defined because of the relation dV ◦ dH = −dH ◦ dV between vertical
and horizontal differentials. In terms of this mapping the Euler–Lagrange equations for a section
s : X → Y of the configurational bundle π takes the intrinsic form(
j∞s
)
iξE(Lη) = 0, ∀ ξ ∈ X
(
J∞(π)
)
.
Last row and the last column in the bicomplex (4.1) form the Euler–Lagrange sequence where
E is the Euler–Lagrange mapping
0→ R→ Ω0,0 dH−−→ Ω1,0 dH−−→ · · · Ωn−1,0 dH−−→ Ωn.0 E−→ F1 δV−→ F2 δV−→ F3 δV−→ · · · .
We recall (see [1, 12]) that the interior Euler (homotopy) operator
I : Ωn,s(J∞(π))→ Fs ⊂ Ωn,s(J∞(π))
that has, as its image, the subbundle Fs of functional forms of vertical degree s is defined by
the formula
I(ω) =
1
s
ωi∧
∑
(i,Λ),|Λ|≥0
(−d)Λ(i∂
zi
Λ
ω)
=
1
s
ωi∧
[
i∂iω−dµ(i∂
ziµ
ω)+dµ1µ2(i∂
ziµ1µ2
ω)−· · ·
]
.(4.2)
Proposition 2. Let KC = Πiω
i ∧ η + Fµi ω
i
µ ∧ η be the K-form corresponding to a constitutive
relation C of order k, see (3.2). Then, the balance system (3.1) is equivalent to the requirement
that
j1s∗iξI(KC) = 0 for all ξ ∈ X
(
Jk(π)
)
. (4.3)
6 S. Preston
Proof.
I(KC) = ωi ∧
[
Πiη − dµ(Fµi η)
]
=
[
Πi − dµFµi − F
µ
i λG,µ
]
ωi ∧ η.
Let ξ ∈ X (J1(π)). Apply iξ to the last equality and take the pullback by the mapping j∞s,
s : U → Y being a section of the bundle π over an open subset U ⊂ X. Equation (4.3) takes
the form(
ωi(ξ) ◦ j1s
)
[dµF
µ
i + Fµi λG,µ −Πi] ◦ j∞sη = 0.
Fulfillment of this equality for all (or for many enough) variational vector fields ξ is equivalent
to the statement that s : U → Y is the solution of the balance system (3.1). �
Remark 2. The result of Proposition 1 shows that a balance system (3.1) is the most general
form of PDEs system that can be obtained from a form of the type (3.2) using the infinitesimal
variational equation (4.3) in the formalism of variational bicomplex. Similar result for the
systems of PDEs starting from an arbitrary Lepage form is obtained in [15].
4.1 Higher order balance systems
Balance systems (3.1) have found their natural place in the variational bicomplex (4.1). Applying
the mapping I to the formKC , contracting it with a variation vector field and taking the pullback
by a 2-jet of a section we restore the balance system (3.1).
Thus, it seems natural to inquire, what higher order construction corresponds to the form K
and which system of PDEs corresponds to this construction. It may lead to the definition of
a canonical form of higher order system of balance equations.
Consider an arbitrary (n, 1)-form on the J∞(π)
K =
∑
|Λ|≥0
FΛ
i ω
i
Λ
∧ η, (4.4)
where FΛ
i ∈ C∞(J∞(π)).
Calculate the functional form I(K) assuming that when σ = 0, dσ is the identity operator.
We get
I(K) = ωi ∧
∑
|Σ|≥0
(−1)|Σ|dΣ
(
i∂
zi
Σ
K
) = ωi ∧
∑
|Σ|≥0
(−1)|Σ|dΣ
(
FΣ
i η
)
= ωi ∧
∑
|Σ|≥0
(−1)|Σ|dΣ
(
FΣ
i |G|
) 1
|G|
η
.
Forming the interior derivative of obtained (n, 1)-form with a vector field ξ ∈ X (J∞(π)) and
taking pullback of obtained horizontal n-form by the ∞-jet j∞s of a section s : U → Y we get
the system of equations for section s(
j∞s
)∗ ∑
|Σ|≥0
(−1)|Σ|ωi(ξ)dΣ
(
FΣ
i |G|
) 1
|G|
η = 0, i = 1, . . . ,m.
Role of the source term in these m equations is played by the functions F 0
i .
Thus, we have proved the following
Balance Systems and the Variational Bicomplex 7
Proposition 3. For a (n, 1)-form K of the form (4.4) on the infinite jet space J∞(π) and
a section s : U → Y where U ⊂ X is an open subset of X, the following statements are
equivalent
1) for all (or, what is equivalent, for m linearly independent) vector fields ξj on J∞(π) such
that det(ωi(ξj)) 6= 0,
j1s∗iξI(K) = 0,
2) section s is the solution of the system of “higher order balance equations”∑
|Σ|>0
(−1)|Σ|−1
|G|
dΣ
(
FΣ
i |G|
)
= Fi, i = 1, . . . ,m. (4.5)
Example 1. Consider the form (4.4) with FΛ
i = ∂L
∂ziΛ
for a function L ∈ C∞(J∞(π)). Then, the
system of PDEs (4.5) takes the form
∂L
∂yi
+
∑
|Σ|>0
1
|G|
(−d)Σ
(
∂L
∂ziΣ
|G|
)
= 0, i = 1, . . . ,m.
Here (−d)Σ = (−1)ΣdΣ. For |G| = 1 this system coincide with the higher order Euler–Lagrange
equations, [3, equation (3.27)] or [12, Section 4.1].
5 Vertical splitting of a balance system
5.1 Vertical homotopy operators
Recall the construction of the vertical homotopy operator hr,sV : Ωr,s → Ωr,s−1 defined by I. An-
derson [1]:
hr,sV (ω) =
∫ 1
0
1
t
Φ∗ln(t)(iprRω)dt, (5.1)
where R = yi∂yi is the Liouville (vertical radial) vector field and
prR = yi∂yi + ziµ∂ziµ + · · ·+ ziΛ∂ziΛ
+ · · ·
is its prolongation to J∞(π). The flow of this vector field on J∞(π) is the 1-parameter family
of diffeomorphisms of J∞(π)
Φε(x, y, z) =
(
z; eεy, eεz
)
.
It is proved in [1] that for any ω ∈ Ωr,s,
ω = dV [hr,sV (ω)] + hr,s+1
V (dV ω). (5.2)
In the proof of this formula it was shown [1, equation (4.10)] that the integrand in the for-
mula (5.1) is equal to(
1
t
Φ∗ln(t)(iprRω)
)
[x, y, z] = ts−2(iprRω)[x; ty, tz] = ts−1iprRω[x; ty, tz].
Here ω[x; ty, tz] is the form obtained by evaluating the coefficients of the form ω at the point
[x; ty, tz]. Using this in (5.1) we get
hr,sV (ω) =
∫ 1
0
ts−1iprRω[x; ty, tz]dt. (5.3)
8 S. Preston
5.2 Decomposition of the form KC, quasi-Lagrangian
We will use the formula (5.3) in the cases r = n, s = 2, 1.
Let ω = KC = Πiω
i ∧ η + Fµi ω
i
µ ∧ η ∈ Ωn,1. Then,
iprRω[x; ty, tz] = Πi(x, ty, tz)iprR
(
ωi ∧ η
)
+ Fµi (x, ty, tz)iprR
(
ωiµ ∧ η
)
=
[
Πi(x, ty, tz)y
iη + Fµi (x, ty, tz)ziµ
]
η,
and hn,1V KC [x; y, z] = L̃η, where the coefficient
L̃(x, y, z) =
∫ 1
0
[
yiΠi(x, ty, tz)dt+ ziµF
µ
i (x, ty, tz)
]
dt (5.4)
of the volume form η will be called the “quasi-Lagrangian” of the form KC in the sense of
decomposition (5.2).
Applying the vertical differential to hn,1V KC [x; y, z] we find the “Lagrangian part” of the
form KC
dV h
n,1
V KC [x; y, z] =
∫ 1
0
(
Πk[] + tyiΠi,yk [] + tziµF
µ
i,yk
[]
)
dtωk ∧ η
+
∫ 1
0
(
tyiΠ
i,zjν
[] + F νj [] + tziµF
µ
i,zjν
[]
)
dtωjν ∧ η
+
∑
Λ||Λ|>1
∫ 1
0
(
tyiΠ
i,zjΛ
[] + tziµF
µ
i,zjΛ
[]
)
dtωjΛ ∧ η. (5.5)
To shorten the formulas we will use the sign [] after function saying that the arguments of
this function are (x, tyi, tziµ, . . . , tz
i
Σ).
Let ω = AΛΣ
ij ω
i
Λ ∧ ω
j
Σ ∧ η ∈ Ωn,2 be any (n, 2)-form. Then the integrand in (5.1) is equal to
tiprRω[x, ty, tz] = tiprR
[
AΛΣ
ij (x, ty, tz)ωiΛ ∧ ω
j
Σ ∧ η
]
= tAΛΣ
ij (x, ty, tz)(ziΛω
j
Σ − z
j
Σω
i
Λ) ∧ η.
Using this in (5.3) we get
hn,2V ω =
(∫ 1
0
tAΛΣ
ij [x, ty, tz]dt
)(
ziΛω
j
Σ − z
j
Σω
i
Λ
)
∧ η.
Apply this for ω = dVKC . We have (using the equality dV ω
i
σ = 0),
dVKC = (dV Πi) ∧ ωi ∧ η + (dV F
µ
i ) ∧ ωiµ ∧ η = Π
i,zjΣ
ωjΣ ∧ ω
i ∧ η + Fµ
i,zjΣ
ωjΣ ∧ ω
i
µ ∧ η.
In this formula |Σ| ≥ 0. Combining this result with the previous calculation we get the comple-
mental, “pure non-Lagrangian”, term in the decomposition (5.2) of the form KC :
hn,2V (dVKC)) =
(∫ 1
0
tΠ
i,zjΣ
(x, ty, tz)dt
)(
zjΣω
i − yiωjΣ
)
∧ η
+
(∫ 1
0
tFµ
i,zjΣ
(x, ty, tz)dt
)(
zjΣω
i
µ − ziµω
j
Σ
)
∧ η
=
{∫ 1
0
t[(yjΠi,yj [] + zjµΠ
i,zjµ
[] + zjΛΠ
i,zjΛ
[])− (yjΠj,yi []− zjµF
µ
j,yi
[])]dt
}
ωi ∧ η
+
{∫ 1
0
t[−yjΠj,ziµ
[] + yjFµ
i,yj
[] + zjνF
µ
i,zjν
[]− zjνF νj,ziµ [] + zjΛF
µ
i,zjΛ
[]]dt
}
ωiµ ∧ η
Balance Systems and the Variational Bicomplex 9
+
{∫ 1
0
t[−yjΠj,ziΛ
− zjνF νj,ziΛ ]dt
}
ωiΛ ∧ η. (5.6)
In the last expression |Λ| > 1 .
Thus, we get
Proposition 4. Vertical splitting (5.2) of a (n, 1)-form KC = Πiω
i∧η+Fµi ω
i
µ∧η has the form
KC = KC,Lag +KC,nLag, (5.7)
where the Lagrangian part KC,Lag is given by (5.5), while the non-Lagrangian part KC,nLag is
given in (5.6).
In the appendix we present several examples of the Lagrangian components (quasi-Lagrangian
and the corresponding Euler–Lagrange equation) of several well known nonlinear equations
having the form of balance equations (see Table 1).
5.3 Properties of vertical decomposition
Consider the bicomplex (4.1) over a domain W ⊂ Y which is vertically star-shaped and covered
by one fibred chart, see [12]. An important special case of this situation is the case where
π : Y → X is the vector bundle. Then, the vertical complex with the vertical differential dV is
dV -exact ([12, Theorem 5.58] or [1, Chapter 4]): any dV -closed (n, r)-form over W is exact. In
addition to this, vertical homotopy operator is defined in the domain W∞ = π−1
∞0(W ).
Denote by Zn,r the subspace of dV -closed forms in Ωn,r. Introduce the subspace Qn,r =
hn,r+1Zn,r+1 ⊂ Ωn,r. In the decomposition ω = hn,r+1dV ω + dV h
n,rω,
hn,r+1dV ω ∈ Qn,r, dV h
n,rω ∈ Zn,r.
Proposition 5. Locally, in the domain W∞ over a vertically star-shaped and covered by one
fibred chart domain W ⊂ Y the following is true:
1. The decomposition (K-decomposition)
Ωn,r(W∞) = Qn,r(W∞)⊕ Zn,r(W∞) (K)
is direct over the domain W .
2. The presentation of a (n, r)-form ω
ω = hn,r+1dV ω + dV h
n,rω (5.8)
as the sum of terms in the decomposition (K) is unique.
3. The mapping V = hn,r+1 ◦ dV : Ωn,r → Ωn,r is the projector: V 2 = V .
4. The mappings
Zn,r+1 hn,r+1
−−−−→ Qn,r and Qn,r
dV−−→ Zn,r+1
are linear isomorphisms inverse to one another.
5. For ν ∈ Ωn,1, hn,rν ∈ Zn,r−1 iff ν ∈ Qn,r.
6. An (n, 1)-form ω is anti-Lagrangian, ω = hn,r+1dV ω ⇔ hn,rω ∈ π∗Ωn(X).
10 S. Preston
7. Over the domain W∞ the following direct decomposition (F -decomposition) at the level of
functional is valid
F1
(
W∞
)
= I
(
Qn,1
)
⊕ E
(
Ωn,0
)
, (F )
where E(Ωn,0) is the space of Lagrangian functional forms.
Proof. To prove the first statement, it is sufficient to show that the intersection of the two
subspaces in the decomposition (K) is zero. Let ω ∈ Qn,r
⋂
Zn,r(W∞), i.e.
ω = hn,r+1σ = dV λ,
where σ = dV σ
′, σ′ ∈ Ωn,r and λ ∈ Ωn,r−1. Applying dV to this decomposition of ω we get
dV ω = 0 = dV h
n,r+1σ and, therefore, dV h
n,r+1dV σ
′ = 0. Using the decomposition (5.2) for σ′
we have hn,r+1dV σ
′ = σ′ − dV hn,rσ′. Substituting this equality into the previous one, we get
dV h
n,r+1dV σ
′ = dV (σ′ − dV hn,rσ′) = dV σ
′ = 0.
It follows from this that σ′ ∈ Zn,r and that σ = dV σ
′ = 0. Therefore, ω = 0.
The second statement is another form of the first one. The third statement follows from the
second one if we apply V to the decomposition (5.8).
To prove 5th statement let ν ∈ Qn,r, then, ν = hn,r+1dV ν + dV h
n,rν. By unicity of such
presentation, dV h
n,rν = 0. This proves the 5th statement and part of statement 4.
Let σ ∈ Zn,r+1, then σ = dV ν, ν ∈ Qn,r. We have hn,r+1σ = hn,r+1dV ν and
dV h
n,r+1σ = dV h
n,r+1dV ν = dV
(
1− dV hn,r+1
)
ν = dV ν = σ.
Combining this with the proven statement ν = hn,r+1dV ν we finish the proof of forth statement.
Next statement follows from the fact that Ker (dV ) in Ωn,0 coincides with π∗Ωn(X).
To prove the last statement that the sum (F ) is direct we use the fact that the homotopy
operator hV and the horizontal differential dH anticommute: If ω ∈ Q and if I(ω) = I(dV Lη) for
some function L, then ω−dV Lη = dHβ for some β ∈ Ωn−1,1. Applying decomposition (K) to the
form dHβ: dHβ = hn,2dV dHβ+dV h
n,1dHβ, substituting it into the expression ω = dV Lη+dHβ
for ω and using unicity in the decomposition (K) we see that ω = hn,2dV dHβ = dHh
n−1,2dV β.
As a result, I(ω) = 0. �
Remark 3. I am grateful to the the referee for the remark that if the configurational bundle
π : Y → X is vector bundle, decomposition (5.2) and (5.8) are globally defined and depend only
on the structure of the vector bundle π.
Remark 4. Projectors I and V do not commute. It follows from the fact that the forms in the
image of I are of the type Piω
i ∧ η while the forms in the image of V have also terms AΛ
i ω
i
Λ ∧ η
with |Λ > 0.
Remark 5. It would be interesting to know, if the restriction of hn,rV to Qn,r is the monomor-
phism? Is it onto the Zn,r−1?
5.4 (n, 1)-forms K with zero Lagrangian part
Here we study the (n, 1)-forms KC which have zero Lagrangian part.
We will be using the following simple lemma.
Balance Systems and the Variational Bicomplex 11
Lemma 1. Let f(ui) be a function of m variables ui, analytical in a connected domain D ⊂ Rm
containing the origin 0. Let s ∈ Z be an integer number. If∫ 1
0
tsf(tu)dt = c− const (respectively zero)
in an open subset O ⊂ D containing the origin, then f(u) = (s+ 1)c− const (respectively zero)
for all u ∈ D.
Proof. Present function f by its Taylor series centered at the origin 0;
f(u) =
∑
I,|I|≥0
cIu
I , (5.9)
where uI = u1 i1 · · ·um im is the monom corresponding to the ordered multiindex I. This Taylor
series absolutely converges in some disc ‖u‖ < R, where R > 0. Rearrange this series collecting
terms cIu
I with |I| = k:
f(u) =
∞∑
k=0
u{k},
where u{k} is the homogeneous part of series (5.9) of degree k. This series also absolutely
converges in the disc ‖u‖ < R. We have, now
f(tu) =
∞∑
k=0
u{k}tk.
As a result,∫ 1
0
tsf(tu)dt =
∞∑
k=0
u{k}
∫ 1
0
ts+kdt =
∞∑
k=0
1
k + s+ 1
u{k} =
∑
I,|I|≥0
1
|I|+ s+ 1
cIu
I . (5.10)
Last series has the same radius of convergence as the series (5.9). Under the condition of
lemma, series (5.10) equal constant c (respectively zero) for all u in the disc ‖u‖ < R. This
series presents an analytical function and is equal to the constant (respectively zero) in the disc
‖u‖ < R. Therefore, series (5.10) have all but the zeroth (respectively all) Taylor coefficients
equal zero. Then, the same is true for the series (5.9) for f(u). Being equal constant (being
identically zero) in an open subset of its domain, function f is the constant (identically zero) in
the whole connected component of its domain containing zero. This proves the lemma. �
Applying this lemma to the quasi-Lagrangian L̃ given by (5.4) presented as the sum of
homogeneous components
L̃ =
∫ 1
0
∞∑
k=0
[
yiΠik(x, y, z) + ziµF
µ
ik(x, y, z)
]
tkdt
we prove the first statement in the following
Proposition 6. Let KC = Πi(x, y
i, ziµ)ωi ∧ η+Fµi (x, yi, ziµ)ωiµ ∧ η be a (n, 1)-form analytical by
vertical variables yi, ziµ in a connected domain D ⊂ Rm+mn containing the origin. Then,
1. The quasi-Lagrangian L̃ corresponding to the form KC is a function of x only (identically
equals zero) if and only if
yiΠi(x, y, z) + ziµF
µ
i (x, y, z) = φ(x) (respectively, = 0). (5.11)
2. If this equality holds, then L̃ = 0.
Proof. Second statement follows from the first and the fact that the analytical function in the
left side of (5.11) vanish when yi = ziµ = 0. �
12 S. Preston
6 Balance systems of zero order
6.1 Balance systems whose quasi-Lagrangian L̃ has zero Euler–Lagrange form
Here we look at the case where Lagrangian part does not enter the functional form of KC (pure
non-Lagrangian case).
We consider here only balance systems of order zero (called the RET systems in continuum
thermodynamics [11]). Thus, the density-fluxes Fµi and the source terms Πi depend on the
fields yi and, possibly, on xµ.
Lagrangian functions L whose (n, 1)-form dV L∧η is of order zero are linear by the derivatives:
L = L0(x, y)+ziµL
µ
i (x, y) and the corresponding Euler–Lagrange systems are quasilinear systems
of the first order(
Lµ
i,yk
− Lµ
k,yi
)
yk,xµ +
(
L0,yi − L
µ
i,xµ
)
= 0, i = 1, . . . ,m. (6.1)
Consider now an arbitrary form K = Πiω
i ∧ η + Fµi ω
i
µ ∧ η of order zero. Quasi-Lagrangian L̃
corresponding to this form is
L̃ =
∫ 1
0
[
ykΠk(x, ty) + zkνF
ν
k (x, ty)
]
dt.
This function depends linearly on the jet variables.
Applying the Euler operator I to the form dV (L̃η) we get
IdV (L̃η) =
(
L̃,yi − dµ
∂L̃
∂ziµ
)
ωi ∧ η.
In the zero order case,
L̃,yi − dµ
∂L̃
∂ziµ
=
∫ 1
0
{
Πi[] + tykΠk,yi [] + tzkνF
ν
k,yi []
}
dt− dµ
∫ 1
0
Fµi []dt
=
∫ 1
0
{
Πi[] + tykΠk,yi [] + tzkνF
ν
k,yi []
}
dt−
∫ 1
0
{
Fµi,µ[] + tzkµF
µ
i,yk
[]
}
dt.
In the homogeneous case, where there is no no explicit dependence of Fµi on xµ, the first term
in the second integral vanishes and we get
L̃,yi − dµ
∂L̃
∂ziµ
=
∫ 1
0
{
Πi[] + tykΠk,yi []
}
dt+
∫ 1
0
{
tzkνF
ν
k,yi []− tz
k
µF
µ
i,yk
[]
}
dt
=
∫ 1
0
{
Πi[] + tykΠk,yi []
}
dt+ zkµ
{(∫ 1
0
Fµk []dt
)
,yi
−
(∫ 1
0
Fµi []dt
)
,yk
}
.
The variables zkµ are independent and are present only as factors in the second term of last line.
so, L̃,yi − dµ ∂L̃
∂ziµ
= 0 if and only if both terms – one containing Πi and the others containing Fµi
all vanish. In particular, we have to have(∫ 1
0
Fµk []dt
)
,yi
=
(∫ 1
0
Fµi []dt
)
,yk
, ∀ i, k.
This condition has the form of mixed derivative test and as a result, Euler–Lagrange equations
for Lagrangian L̃ vanishes if and only if the following conditions holds:∫ 1
0
{
Πi[] + tykΠk,yi []
}
dt =
∂
∂yi
∫ 1
0
ykΠk[]dt = 0,
Balance Systems and the Variational Bicomplex 13
locally, for some functions Gµ(yi),∫ 1
0
Fµk (ty)dt =
∂Gµ
∂yk
, µ = 1, . . . , n, k = 1, . . . ,m.
Using Lemma 1 we show that in the analytical case the first condition is equivalent to the
condition ykΠk(x, y) = c(x) and, due to the analyticity by vertical variables, to the equality
ykΠk(x, y) = 0. More than this, in the analytical case we can rearrange terms of Taylor series
for the functions Fµk and Gµ(y) collecting together monoms yI = y1i1 · · · ym im with the same
value of |I| = k:
Fµk =
∑
p
Fµkp, Gµ =
∞∑
l=0
Gµl . (6.2)
Substituting this into the second condition and using Lemma 1, we write it in the form
∞∑
l=0
∂Gµl
∂yk
=
∑
p
Fµkp(y)
∫ 1
0
tpdt =
∑
p
1
p+ 1
Fµkp(y).
Equating homogeneous terms of the same order in both parts we get
1
p+ 1
Fµkp(y) =
∂Gµp+1
∂yk
, p = 0, 1, . . . .
Multiplying by (p+ 1) and taking summation by p we get the formal equality
Fµk =
∞∑
p=0
Fµkp(y) =
∞∑
p=0
(
(p+ 1)Gµp+1
)
,yk
=
∂
∂yk
∞∑
p=0
(p+ 1)Gµp+1
.
It is easy to see that the series on the right side has the same radius of convergence as the
corresponding series in (6.2). Denote obtained analytical function by
G̃µ(y) =
∞∑
p=0
(p+ 1)Gµp+1(y).
As a result, we have proved the following
Proposition 7. Let the balance system (3.1) be analytical and of order zero (RET case). Then
the Lagrangian part of this balance system in the functional form splitting (F ) vanishes, or, in
other terms, I(KC) ∈ I(Qn,1) if and only if the following conditions are fulfilled:
1) ykΠk(x, y) = c− const,
2) Godunov condition (see below Definition 1): the form Fµi ω
i ∧ ηµ is vertically closed, i.e.
locally, Fµi ω
i ∧ ηµ = dV (Gµηµ) for some analytical functions G̃µ(yi), µ = 1, . . . , n. As
a result,
Fµk (y) =
∂G̃µ
∂yk
, µ = 1, . . . , n, k = 1, . . . ,m.
Remark 6. Notice that the Godunov condition has the form dV F
µ
i ω
i ∧ ηµ = 0, is covariant
and, therefore, globally well defined. The second condition: yiΠi = 0 can be written, using the
Liouville vector field R = yi∂yi , in the form iRKC = 0. Thus, this condition is globally well
defined on the vector bundles π : Y → X where the transition transformations are linear on the
fibers of π.
14 S. Preston
6.2 Systems of Godunov and the decomposition of the (n, 1)-forms
of order zero
In 1961, S. Godunov [5] defined an interesting class of balance systems of the form
µ=3∑
µ=0
d
dxµ
(
∂Gµ
∂yi
)
= 0, i = 1, . . . ,m, (6.3)
where Gµ(y) are some functions of the dynamical fields yi. These systems are generalizations of
symmetrical hyperbolic systems of Friedrichs. Later on, S. Godunov and his collaborators used
these systems as a starting point for constructing Galilean invariant and thermodynamically
compatible models of a large class of physical systems [6, 7]. This was done by adding terms
depending on the derivatives ziµ, to the fluxes FAi , A = 1, 2, 3 and introducing the source
terms Πi. At the same time, G. Boillat and T. Ruggeri proved that the Godunov systems (6.3)
present the canonical form of a RET (zero order) balance system with the entropy balance if the
entropy density is convex (condition guaranteeing the thermodynamical stability) [11, Chapter 3,
Section 2].
Thus, it seems natural to introduce
Definition 1. A balance system (3.1) of order zero and the corresponding (n, 1)-form K will
be called Godunov type system if the density-flux part of the system has the form
Fµi =
∂Gµ
∂yi
for a functions Gµ ∈ C∞(Jk(π)).
Thus, the result of Proposition 7 can be formulated as follows:
Corollary 1. An analytical balance system of zero order is pure non-Lagrangian if and only if
it is of Godunov type and its source part Πiω
i ∧ η satisfies to the condition
ykΠk(x, y) = 0.
Next we project this result to the complex of functional forms (most right column in (4.1)),
where δV = I◦dV is the induced vertical differential. Using the vertical splitting (F ) of functional
forms (see also [1, Chapter 4A]) we get the following
Corollary 2. Let the domain W ⊂ Y satisfies to the conditions of Proposition 5. Then, any
analytical functional form K ∈ F1 of order zero in the domain W is the sum
I(K) = GK + EL̃K
of the functional forms generated by: form of Godunov type
GK =
[
(Πi − L̃K,yi)ωi + (Fµi − L̃K,ziµ)ωiµ
]
∧ η
and the Euler form
EL̃ = dV L̃K ∧ η
for the quasi-Lagrangian L̃K . This decomposition is unique (in the domain W ).
Proof. Present the (n, 1)-form K as the sum K = K1 + dV (L̃Kη) as in (5.7). Component
K1 ∈ Qn,1 in this representation has the property that L̃K1 ∈ C∞(X) and, therefore, ELK1
= 0.
From the previous corollary it follows that K1 is of Godunov type. Then, since I2 = I, K =
I(K) = I(K1) + EL̃K . Now we rename I(K1) = GK and the proof is complete. �
Balance Systems and the Variational Bicomplex 15
Comparing systems of equations corresponding to the Lagrangian component dV (L̃η) and
the Godunov component of a (n, 1)-form K = Πiω
i ∧ η + Fµi ω
i
µ ∧ η with Πi, F
µ
i ∈ C∞(Y ), we
see that the tensor Aµij of the principal parts Aµij(x, y)∂
yj
,xµ
(of i-th equation) is symmetrical
(by ij) for the Godunov component of the balance system and anti-symmetrical (see (6.1)) for
the Lagrangian component.
Let now K = Πi(x, y)ωi ∧ η + Fµi (x, y)ωiµ ∧ η be a (n, 1)-form of order zero of the type
considered here. Let L̃ = L̃0(x, y) + ziµL̃
µ
i (x, y) be the corresponding quasi-Lagrangian. Then
L̃0
i =
∫ 1
0 F
0
i (x, ty)dt. Godunov component GK of the form K has the functions F̃µi = Fµi −
L̃K,ziµ = Fµi −
∫ 1
0 F
µ
i (x, ty)dt as its densities (µ = 0) and fluxes (µ = 1, 2, 3). Matrices
F̃µ
i,yj
= Fµ
i,yj
−
∫ 1
0
tFµ
i,yj
(x, ty)dt
are symmetric by (i, j) (see Definition 1). As a result, the system of balance equations corre-
sponding to the form GK is the first order system of PDEs Mµ
ij(x, y)yj,µ = bi(x, y) with symmet-
rical matrices Mµ
ij . If the matrix M0
ij = F̃ 0
i,yj
is positive definite, this system is the symmetrical
hyperbolic by Friedrichs [11]. Thus, we get
Corollary 3. Let K = Πi(x, y)ωi ∧ η + Fµi (x, y)ωiµ ∧ η be an analytical by yi (n, 1)-form of
order zero and let the symmetrical matrix M0
ij = F̃ 0
i,yj
is positive definite. Then the system of
balance equations corresponding to the Godunov component GK of the form K is the symmetrical
hyperbolic system by Friedrichs.
We finish this section with two examples of calculation of quasi-Lagrangian for two balance
systems – the system of elasticity equations for a hyperalastic body and the reduced system of
equations of 2-dimensional ideal plasticity.
6.3 Example: hyperelasticity
Consider the system of equations of hyperelasticity [18, Section 1.6].
Let M3 be a material manifold – a connected open 3-dimensional manifold endowed with
the (reference) Riemannian metric g0. Local coordinates in M will be denoted by XI , I =
1, 2, 3, time t by X0. A deformation history of the body M is a time dependent diffeomorphic
embedding φ : M → E3 of M into the Euclidian 3-dimensional (physical) space E3 endowed
with the metric h. Local coordinates in E3 will be denoted by xi, i = 1, 2, 3. As the dynamical
variables of Elasticity we choose the velocity vector field over φ: v = ∂φ
∂X0 and the deformation
gradient F, F iI = ∂φi
∂XI – (1, 1)-tensor over the mapping φ.
Constitutive properties of the body are determined by the strain energy function w = w(F iµ).
Strain energy w depends on the material metric g0 and the physical Euclidian metric h [10].
Elasticity equations have the form of the balance system of order zero, where the first equation
is the linear momentum balance law and the second one – compatibility condition,
∂X0vi + ∂XI
(
∂w
∂F iI
)
= bi(X), ∂X0F iI − ∂XI
(
δikv
k
)
= 0. (6.4)
Dynamical fields are yi = vi, i = 1, 2, 3 and yaA = F aA, a,A = 1, 2, 3. Jet variables will be
denoted by ziµ, i = 1, 2, 3, µ = 0, 1, 2, 3, corresponding to the derivatives of velocity components
and zaAµ , a,A = 1, 2, 3, µ = 0, 1, 2, 3, corresponding to the derivatives of components of the
deformation gradient. Basic contact forms in the second jet bundle J2(π) are
ωi = dvi − ziνdXν , ωaA = dF aA − zaAν dXν ,
16 S. Preston
ωiµ = dvi,µ − ziµνdXν , ωaAµ = dF aA,µ − zaA,µνdXν .
The form KC in this case is
KC = (biω
i ∧ η) +
(
viω
i
0 + w,FaAω
aA
)
∧ η +
(
F aAω
aA
0 − δBAvAωaAB
)
∧ η.
To calculate the quasi-Lagrangian L̃ =
∫ 1
0 [yiΠi[] + zi0F
0
i [] + ziAF
A
i []]dt we write∫ 1
0
[yiΠi]dt = vibi,
which has the meaning of mechanical power of the bulk force bidx
i (we assume that bj(X) are
functions of X only).
Next, using the compatibility equation, we get for the terms corresponding to the elasticity
equations∫ 1
0
(
zi0F
0
i [] + ziAF
A
i []
)
dt =
∫ 1
0
(
zi0tvi + ziAw,F iA
[]
)
dt ∼ 1
2
vi∂tv
i + vi,xA
∫ 1
0
w,F iA
[]dt
= ∂X0
(
‖v‖2
2
)
+ (∂X0F iA)∂F iA
∫ 1
0
t−1w[]dt = ∂X0
(
‖v‖2
2
+
∫ 1
0
t−1w[]dt
)
.
We recognize the time derivative of the energy density of elastic body (sum of kinetic energy
and strain energy). Finally, using F aB,0 = va
,XB , we get∫ 1
0
[
zaB0 F 0
aB + zaBA FAaB
]
dt =
∫ 1
0
[
zaB0 F aBt+ zaBA δABv
at
]
dt ∼ 1
2
(
F aB,0F
a
B + F aB,XBv
a
)
=
1
2
(
va,BF
a
B + vaF aB,xB
)
=
1
2
∂XB
(
vaF aB
)
.
Combining we get (partial derivatives here coincide with the total derivatives)
L̃ = vibi + dX0
(
‖v‖2
2
+
∫ 1
0
t−1w[]dt
)
+ dXB
(
1
2
vaF aB
)
.
Thus, the quasi-Lagrangian is the sum of the trivial Lagrangian and the source term that
generates the right side in the elasticity equation (6.4). Notice that the trivial part of L̃ has the
form of the left side of a balance law similar to the energy balance for the elastic body.
6.4 Example: 2-dimensional ideal plasticity
System of equations described the plane strained state of the medium with von Mises conditions
can be transformed, by change of variables (see [17]), to the system of balance equations
u,ξ +
1
2
v = 0, v,η +
1
2
u = 0 (6.5)
for the functions y1 = u, y2 = v of variables x0 = ξ, x1 = η. For this balance system, F 0
1 = u,
F 1
1 = 0, F 0
2 = 0, F 1
2 = v, Π1 = −1
2v, Π2 = −1
2u.
The quasi-Lagrangian of this system is
L̃ =
1
2
[
∂ξ
u2
2
+ ∂η
v2
2
]
− uv.
Thus, in this case the quasi-Lagrangian is also sum of total divergence and of the expression
producing the source terms in the balance system (6.5).
Balance Systems and the Variational Bicomplex 17
7 Conclusion
This works shows that the general systems of m balance equations (3.1) for m dynamical fields
yi represent the natural type of systems of PDEs appearing in the framework of variational
bicomplex. Any such system is associated with the (n, 1)-form KC = Πiω
i ∧ η + Fµi ω
i
µ ∧ η ∈
Ωn,1(J∞(π)). It is proven that the vertical homotopy formula delivers, in appropriate domains
or, for vector bundles, globally, the unique representation of KC as the sum of the Lagrangian
term and the “pure non-Lagrangian” form. The corresponding splitting of the functional forms
follows from the decomposition of (n, 1)-forms. It is shown that for zero order forms KC , whose
quasi-Lagrangian L̃ is trivial (I(dV L̃η) = 0), the functional form I(KC) is of the type introduced
by S. Godunov in 1961 [5], and, later on, identified as the canonical (symmetrical hyperbolic)
form of balance system in the Rational Irreducible Thermodynamics, [11].
In the continuation of this work we will study the decomposition (5.7) of the functional forms
into the Lagrangian and non-Lagrangian parts for the balance systems of order one. Another
natural direction of future work is to study the question (suggested by the referee): is it possible
to obtain a global version of the splitting (F )? Finally, it would be interesting to extend results
obtained for the case of analytical constitutive relations (density/flux and source terms Fµi , Πi)
to the smooth (C∞) constitutive relations.
A Examples of vertical decomposition
In this appendix we present a few examples of the Lagrangian component of several well known
nonlinear equations having the form of balance equations. As these examples show, the terms
in second order spacial derivatives arrive from the Lagrangian component. The source terms in
examples of Subsections 6.3, 6.4 also come from the Lagrangian component. The same is true
for the source term in the last equation below. It is possible that observations of this kind will
be useful for the construction of thermodynamically compatible balance systems for different
physical systems (comp. [6, 7]).
Table 1.
Name Equation quasi-Lagrangian L̃ Lagr. part of eq.
conser-
vation law
∂tu+ ∂xC(u) = 0 dt
(
u2
4
)
+ dx
(∫ 1
0
∫ u
0
C(ty)dydt
)
0 = 0
KdV
equation
∂tu+ ∂x
(
3u2 + uxx
)
= 0 dt
(
u2
4
)
+ dx
(
u2
,x
4
+ u3
3
)
0 = 0
Burgers
equation
∂tu− ∂x
(
u2
2
+ ux
)
= 0 dt
u2
4
+ dx
(
−u
3
18
)
− u2
,x
2
uxx = 0
Filtration
equation [2]
(u−∆u),t − (u,xA),xA = 0
(
u2
4
+ ‖∇u‖2
4
)
,t
−
(
u,tu,xA
2
)
,xA
+ ‖∇u‖2
2
∆u = 0
cubic Schrö-
dinger equation
iu,t + ∆u+ α|u|2u = 0 dt(iu
2) + 1
2
‖∇u‖2 − α
4
u|u|2 ∆u+ α
4
|u|2u = 0
Acknowledgments
I would like to thank the second referee for his recommendations and comments. They allowed
me to correct and/or clarify the formulations of some results and their proofs.
References
[1] Anderson I., The variational bicomplex, Preprint, Utah State University, 2003.
[2] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V.,
Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential
equations of mathematical physics, American Mathematical Society, Providence, RI, 1999.
18 S. Preston
[3] Campos C., de Leon M., Diego D., Vankerschaver J. Unambiguous formalism for higher-order Lagrangian
field theories, arXiv:0906.0389.
[4] Fatibene L., Francaviglia M., Natural and gauge natural formalism for classical field theories. A geometric
perspective including spinors and gauge theories, Kluwer Academic Publishers, Dordrecht, 2003.
[5] Godunov S.K., An interesting class of quasilinear systems, Soviet Math. Dokl. 2 (1961), 947–948.
[6] Godunov S.K., Gordienko V.M., The simplest galilean-invariant and thermodynamically consistant conser-
vative laws, J. Appl. Mech. Tech. Phys. 43 (2002), 1–12.
[7] Godunov S.K., Structure of thermodynamically compatible systems, Appendix to Godunov S.K., Romen-
skii E.I., Elements of continuum mechanics and conservation laws, Kluwer Academic/Plenum Publishers,
New York, 2003.
[8] Kolář I., Michor P., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
[9] Krupka D., Global variational theory in fibred spaces, in Handbook of Global Analysis, Editors D. Krupka
and D. Saunders, Elsevier Sci. B.V., Amsterdam, 2008, 773–836.
[10] Marsden J.E., Hughes T.J.R., Mathematical foundations of elasticity, Dover Publications, Inc., New York,
1994.
[11] Müller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy,
Vol. 37, Springer-Verlag, New York, 1998.
[12] Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics,
Vol. 107, Springer-Verlag, New York, 1993.
[13] Preston S., Variational theory of balance systems, in Proceedings of Conf. “Differential Geometry and its
Applications in Honour of Leonhard Euler” (August 27–31, 2007, Olomouc), Editors O. Kowalski, D. Krupka,
O. Krupková and J. Slovák, World Sci. Publ., Hackensack, NJ, 2008, 675–688, arXiv:0806.4636.
[14] Preston S., Geometrical theory of balance systems, Int. J. Geom. Methods Mod. Phys. 7 (2010), 745–795.
[15] Preston S., Forms of Lepage type and the balance systems, Differential Geom. Appl., to appear.
[16] Saunders D., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142,
Cambridge University Press, Cambridge, 1989.
[17] Senashov S.I., Vinogradov A.M., Symmetries and conservation laws of 2-dimensional ideal plasticity, Proc.
Edinburgh Math. Soc. (2) 31 (1988), 415–439.
[18] Serre D., Systems of conservation laws. I. Hyperbolicity, entropies, shock waves, Cambridge University Press,
Cambridge, 1999.
[19] Sieniutycz S., Conservation laws in variational thermohydrodynamics, Mathematics and its Applications,
Vol. 279, Kluwer Academic Publishers Group, Dordrecht, 1994.
[20] Sieniutycz S., Farkas S., Variational and extremum principles in macroscopic systems, Elsevier, Amsterdam,
2005.
http://arxiv.org/abs/0906.0389
http://dx.doi.org/10.1023/A:1013981706675
http://dx.doi.org/10.1016/B978-044452833-9.50016-4
http://arxiv.org/abs/0806.4636
http://dx.doi.org/10.1142/S0219887810004543
http://dx.doi.org/10.1016/j.difgeo.2011.04.026
http://dx.doi.org/10.1017/CBO9780511526411
http://dx.doi.org/10.1017/S0013091500006817
http://dx.doi.org/10.1017/S0013091500006817
http://dx.doi.org/10.1017/CBO9780511612374
1 Introduction
2 Settings and notations
3 Balance systems, form KC and the Helmholtz coindition
4 Balance systems and the variational bicomplex
4.1 Higher order balance systems
5 Vertical splitting of a balance system
5.1 Vertical homotopy operators
5.2 Decomposition of the form KC, quasi-Lagrangian
5.3 Properties of vertical decomposition
5.4 (n,1)-forms K with zero Lagrangian part
6 Balance systems of zero order
6.1 Balance systems whose quasi-Lagrangian has zero Euler-Lagrange form
6.2 Systems of Godunov and the decomposition of the (n,1)-forms of order zero
6.3 Example: hyperelasticity
6.4 Example: 2-dimensional ideal plasticity
7 Conclusion
A Examples of vertical decomposition
References
|