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variational bicomplex

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Variational calculus

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Contents

Idea

For XX a (spacetime) manifold and EXE \to X a bundle (in physics called the field bundle) with jet bundle Jet(E)XJet(E) \to X, the variational bicomplex is essentially the de Rham complex (Ω (Jet(E)),d)(\Omega^\bullet(Jet( E)),\mathbf{d}) of Jet(E)Jet(E) with differential forms Ω n(Jet(E))= h+v=nΩ h,v(E)\Omega^n(Jet(E)) = \bigoplus_{h+v=n} \Omega^{h,v}(E) bigraded by horizontal degree hh (with respect to XX) and vertical degree vv (along the fibers of j Ej_\infty E)). Accordingly the differential decomposes as

d=d+δ, \mathbf{d} = d + \delta \,,

where d\mathbf{d} is the de Rham differential on Jet(E)Jet(E), dd is called the horizontal differential and δ\delta is called the vertical differential.

With EXE \to X thought of as a field bundle over spacetime/worldvolume, then dd is a measure for how quantities change over spacetime, while δ\delta is the variational differential that measures how quantities change as the field configurations are varied.

Accordingly, much of classical mechanics and classical field theory on XX is formalized in terms of the variational bicomplex. For instance

  • a field configuration is a section of EE;

  • a Lagrangian is an element LΩ n,0(E)L \in \Omega^{n,0}(E);

  • a local action functional is a map

    S:Γ(E) S : \Gamma(E) \to \mathbb{R}

    of the form

    S(ϕ)= XL(j ϕ), S(\phi) = \int_X L(j^\infty \phi) \,,
  • the Euler-Lagrange equation is

    E(L):=δLmodimd=0 E(L) := \delta L \mod im d = 0
  • the covariant phase space is the locus

    {ϕΓ(E)|E(L)(j ϕ)=0} \{ \phi \in \Gamma(E) | E(L)(j^\infty \phi) = 0 \}
  • a conserved current is an element ηΩ n1,0(E)\eta\in \Omega^{n-1,0}(E) that is horizontally closed on the covariant phase space

    dη=0modE(L) d \eta = 0 \mod E(L)
  • a symmetry is an evolutionary vector field vv such that

    v(L)=0modimd v(L) = 0 \mod im d
  • Noether's theorem asserts that every symmetry induces a conserved current.

Definition

Let XX be a smooth manifold and p:EXp : E \to X some smooth bundle over XX. Write Jet(E)XJet(E) \to X for the corresponding jet bundle.

The bicomplex

The spaces of sections Γ(E)\Gamma(E) and Γ(Jet(E))\Gamma(Jet(E)) canonically inherit a generalized smooth structure that makes them diffeological spaces: we have a pullback diagram of diffeological spaces

Γ(E) * id [X,E] p * [X,X]. \array{ \Gamma(E) & \longrightarrow & * \\ \downarrow && \downarrow^{id} \\ [X,E] &\stackrel{p_*}{\longrightarrow}& [X,X] } \,.

This induces the evaluation map

X×Γ(E)E. X \times \Gamma(E) \to E \,.

and composed with the jet prolongation

j :Γ(E)Γ(Jet(E)) j^\infty : \Gamma(E) \to \Gamma(Jet(E))

it yields a smooth map (homomorphism of diffeological spaces)

(1)e :X×Γ(E)(id,j )X×Γ(Jet(E))evJet(E). e_\infty : X \times \Gamma(E) \stackrel{(id,j^\infty)}{\to} X \times \Gamma(Jet(E)) \stackrel{ev}{\to} Jet(E) \,.

Write

Ω ,(X×Γ(E)) \Omega^{\bullet, \bullet}(X \times \Gamma(E))

for the cochain complex of smooth differential forms on the product X×Γ(E)X \times \Gamma(E), bigraded with respect to the differentials on the two factors

dd+δ, \mathbf{d} \coloneqq d + \delta \,,

where the d\mathbf{d}, dd and δ\delta, are the de Rham differentials of X×Γ(E)X\times\Gamma(E), XX and Γ(E)\Gamma(E), respectively.

Definition

The variational bicomplex of EXE \to X is the sub–bi-complex of Ω ,(X×Γ(E))\Omega^{\bullet, \bullet}(X \times \Gamma(E)) that is the image of the pullback of forms along the map e e_\infty (1):

e *:Ω (Jet(E))Ω (X×Γ(E)). e_\infty^* : \Omega^{\bullet}(Jet(E)) \to \Omega^\bullet(X \times \Gamma(E)) \,.

We write

Ω loc ,im(e *) \Omega^{\bullet, \bullet}_{loc} \coloneqq im (e_\infty^*)

and speak of the bicomplex of local forms on sections on EE.

The bicomplex structure on Ω loc ,\Omega^{\bullet, \bullet}_{loc} is attributed in (Olver 86) to (Takens 79). The above formulation as a sub-bicomplex of the evident bicomplex of forms on X×Γ(E)X \times \Gamma(E) is due to (Zuckerman 87, p. 5).

More on the horizontal differential complex

Remark

In terms of a coordinate chart (x i,u α,u i α,u ij α,)(x^i, u^\alpha,u^\alpha_i,u^\alpha_{i j},\cdots) of EE covering a coordinate chart (X i)(X^i) of XX, the action of the horizontal differential on functions fC (Jet(E))f \in C^\infty(Jet(E)) is given by the formula for the total derivative operation, but with concrete differentials substituted by the respective jet coordinates:

df= i(fx i+fu αu i α+ jfu j αu ji α+)dx i. d f = \sum_i \left( \frac{\partial f}{\partial x^i} + \frac{\partial f}{\partial u^\alpha}u^\alpha_i + \sum_j \frac{\partial f}{\partial u^\alpha_j} u^\alpha_{j i} + \cdots \right) d x^i \,.

(For instance (Anderson 89, p. 10).)

More abstractly, the horizontal differential is characterized as follows:

Proposition

The horizontal differential takes horizontal forms to horizontal forms, and for all sections ϕΓ(E)\phi \in \Gamma(E) it respects pullback of differential forms along the jet prolongation j ϕΓ(Jet(E))j_\infty \phi \in \Gamma(Jet(E))

(j ϕ) *d=d(j ϕ) * (j_\infty \phi)^\ast \circ d = d \circ (j^\infty \phi)^\ast

(where on the right we have the ordinary de Rham differential on the base space).

More abstractly, the horizontal complex may be understood in terms of differential operators and the jet comonad as follows.

Remark

A horizontal differential nn-form α\alpha on Jet(E)XJet(E) \to X is equivalently a homomorphism of bundles over XX

α:Jet(E) nT *X \alpha \colon Jet(E) \longrightarrow \wedge^n T^\ast X

from the jet bundle Jet(E)Jet(E) to the exterior bundle nT *X\wedge^n T^\ast X. This in turn is, by the discussion there, equivalently a differential operator α:E nT *X\alpha \colon E \to \wedge^n T^\ast X.

Now of course also the de Rham differential d Xd_X on XX is a differential operator nT *X nT *X\wedge^n T^\ast X \to \wedge^n T^\ast X. In view of this, the horizontal differential of the variational bicomplex is just the composition operation of differential operators, with horizontal forms regarded as differential operators as above.

By the fact that differential operators are the co-Kleisli morphisms of the Jet comonad, this means that the horizontal differential is

d Hα:Jet(F)Jet(Jet(F))Jet(α)Jet( nT *X)d˜ X nT *X. d_H \alpha \colon Jet(F) \longrightarrow Jet(Jet(F)) \stackrel{Jet(\alpha)}{\longrightarrow} Jet(\wedge^n T^\ast X) \stackrel{\tilde d_X}{\longrightarrow} \wedge^n T^\ast X \,.

(e.g. Krasil’shchik-Verbovertsky 98, around def. 3.27, Krasil’shchik-Vinogradov 99, ch 4, around def. 1.8)

Evolutionary vector fields

Vector fields on J EJ^\infty E also split into a direct sum of vertical and horizontal ones, respectively being annihilated by contraction with any horizontal 11-forms or with any vertical 11-forms, 𝔛(J E)=𝔛 H(J E)𝔛 V(J E)\mathfrak{X}(J^\infty E) = \mathfrak{X}_H(J^\infty E) \oplus \mathfrak{X}_V(J^\infty E). A special kind of vertical vector field v𝔛 V(J E)v \in \mathfrak{X}_V(J^\infty E) is called an evolutionary vector field provided it satisfies vd=d v\mathcal{L}_v d = d \mathcal{L}_v and v=ι vδ+δι v\mathcal{L}_v = \iota_v \delta + \delta \iota_v, we denote the subspace of evolutionary vector fields as 𝔛 ev(J E)𝔛 V(J E)\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E).

Properties

Horizontal, vertical, and total cohomology

Let EXE \to X be a smooth fiber bundle over a base smooth manifold XX of dimension n.n. Write J EXJ^\infty E \to X for the jet bundle of EXE\to X.

Write

s(J E)I(Ω n,s(J E)) \mathcal{F}^s(J^\infty E) \coloneqq I (\Omega^{n,s}(J^\infty E))

for the projection of (n,s)(n,s)-forms to the image of the “interior Euler operator” (Anderson 89, p. 21 (50/318)).

Proposition

The cochain cohomology of the Euler-Lagrange complex

0Ω 0,0(J E)d HΩ 1,0(J E)d Hd HΩ n,0(J E)E 1(J E)δ V 2(J E)δ V 0 \to \mathbb{R} \to \Omega^{0,0}(J^\infty E) \stackrel{d_H}{\to} \Omega^{1,0}(J^\infty E) \stackrel{d_H}{\to} \cdots \stackrel{d_H}{\to} \Omega^{n,0}(J^\infty E) \stackrel{E}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_V}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots

is isomorphic to the de Rham cohomology of the total space EE of the given fiber bundle.

(Anderson 89, theorem 5.9).

The fundamental variational formula

Definition

A source form is an element α\alpha in Ω loc n,1\Omega^{n,1}_{loc} such that

α ϕ(δϕ) \alpha_\phi(\delta \phi)

depends only on the 0-jet of δϕ\delta \phi.

Proposition

Let LΩ loc n,0L \in \Omega^{n,0}_{loc}.

Then there is a unique source form E(L)E(L) such that

δL=E(L)dΘ. \delta L = E(L) - d \Theta \,.

Moreover

  • E(L)E(L) is independent of changes of LL by dd-exact terms:

    E(L)=E(L+dK). E(L) = E(L + d K) \,.
  • Θ\Theta is unique up to dd-exact terms.

This is (Zuckerman 87, theorem 3).

Here EE is the Euler-Lagrange operator .

Definition

Write

Ω=δΘ. \Omega = \delta \Theta \,.
Remark

By prop. 3 have

dΩ=δE(L). d \Omega = -\delta E(L) \,.
Proposition

δE\delta E vanishes when restricted to vertical tangent vectors based in covariant phase space (but not necessarily tangential to it).

δE(L)| T E(L)=0Γ(E)=0. \delta E(L) |_{T_{E(L) = 0} \Gamma(E)} = 0 \,.

This is ([Zuckerman 87, lemma 8]).

Presymplectic covariant phase space

Corollary

The form Ω\Omega is a conserved current.

Proof

By remark 3 and prop. 4.

Definition

For ΣX\Sigma \subset X a compact closed submanifold of dimension n1n-1, one says that

ω:= ΣΩΩ loc 0,2 \omega := \int_\Sigma \Omega \in \Omega^{0,2}_{loc}

is the presymplectic structure on covariant phase space relative to Σ\Sigma.

Proposition

The 2-form ω\omega is indeed closed

δω=0 \delta \omega = 0

and in fact exact:

θ:= ΣΘ \theta := \int_\Sigma \Theta

is its presymplectic potential .

δθ=ω. \delta \theta = \omega \,.

Symmetries

Let LΩ loc n,0L \in \Omega^{n,0}_{loc}.

Definition

An evolutionary vertical vector field v𝔛 ev(J E)v \in \mathfrak{X}_{ev}(J^\infty E) is a symmetry if

v(L)=0modimd. v(L) = 0 \mod im d \,.
Proposition

The presymplectic form ω\omega from def. 4 is annihilated by the Lie derivative of the vector field on Γ(E)\Gamma(E) induced by a symmetry.

This appears as (Zuckerman 87, theorem 13).0

Elementary formalization in differential cohesion

We discuss aspects of an elementary formalization in differential cohesion of the concept of the variational bicomplex .

under construction

Let H\mathbf{H} be a context of cohesion and differential cohesion, with

Choose

  1. an object ΣH\Sigma \in \mathbf{H}, the base space (or spacetime or worldvolume);

  2. an object EH /ΣE \in \mathbf{H}_{/\Sigma}, the field bundle,

  3. an object AStab(H /Σ)ΩH /Σ\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}, the differential coefficients.

Write

  • H /Σ ΣΣ * ΣH\mathbf{H}_{/\Sigma} \stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}} \mathbf{H} for the base change adjoint triple over Σ\Sigma, the étale geometric morphism of the slice (infinity,1)-topos H /Σ\mathbf{H}_{/\Sigma};

  • Γ X Σ:H /ΣH\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H} for the external space of sections functor;

  • i:Σ(Σ)i \colon \Sigma \longrightarrow \Im(\Sigma) for the Σ\Sigma-component of the unit of \Im;

  • Jet Σi *i *Jet_\Sigma \coloneqq i^\ast i_\ast for the induced jet comonad;

  • H /ΣιPDE(H) Σ\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma} for the Eilenberg-Moore category of Jet ΣJet_\Sigma-coalgebras (the objects are differential equations with variables in Σ\Sigma, the morphisms are differential operators between these, preserving solution spaces), manifested as a topos of coalgebras over H\mathbf{H};

    the (non-full) direct image of this geometric morphism is the co-Kleisli category of the jet comonad and so for ϕ:free(E)free(F)\phi \colon free(E) \to free(F) a morphism in PDE(H) ΣPDE(\mathbf{H})_\Sigma, we write f˜:Jet(E)F\tilde f \colon Jet(E) \to F for the corresponding co-Kleisli morphism in H /Σ\mathbf{H}_{/\Sigma};

We record the following simple fact, which holds generally since the jet comonad Jet ΣJet_\Sigma is a right adjoint (to the infinitesimal disk bundle functor), hence preserves terminal objects, and ΣH /Σ\Sigma \in \mathbf{H}_{/\Sigma} is the terminal object:

Proposition

The essentially unique morphism

Jet(Σ)Σ Jet(\Sigma) \stackrel{\simeq}{\longrightarrow} \Sigma

in H /Σ\mathbf{H}_{/\Sigma} in an equivalence.

Definition

The jet prolongation map

j:Γ Σ(E)Γ Σ(Jet(E)) j \colon \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(Jet(E))

is the the Jet functor itself, regarded, in view of prop. 7, as taking sections to sections via

(ΣσE)(ΣJet(Σ)Jet(σ)Jet(F)). (\Sigma \stackrel{\sigma}{\to} E) \;\;\mapsto \;\; \left( \Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet\left(F\right) \right) \,.
Definition

For EH ΣE \in \mathbf{H}_{\Sigma} a bundle over Σ\Sigma, then a horizontal A\mathbf{A}-form on the jet bundle Jet(E)Jet(E) is a morphism in PDE(H) ΣPDE(\mathbf{H})_{\Sigma} of the form

α:ιEιA. \alpha \colon \iota E \to \iota \mathbf{A} \,.

For d:ιEt ΣΣ *AιEt ΣΣ *Ad \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A}\to \iota Et_\Sigma\Sigma^\ast \mathbf{A}' a morphism in PDE(H) Σ\mathbf{PDE}(\mathbf{H})_{\Sigma}, then the induced horizontal differential is the operation of horizontal forms sending α\alpha to the composite

dα:ιEαιAdιA. d \alpha \colon \iota E \stackrel{\alpha}{\longrightarrow} \iota \mathbf{A} \stackrel{d}{\longrightarrow} \iota \mathbf{A}' \,.
Remark

Since all objects in def. 7 are in the co-Kleisli category of the jet comonad, the morphism α\alpha there is equivalently a morphism in H /Σ\mathbf{H}_{/\Sigma} of the form

α˜:Jet(E)A. \tilde \alpha \colon Jet(E) \longrightarrow \mathbf{A} \,.

For the special case that E=ΣE = \Sigma in def. 7, then Jet Σ(Σ)ΣJet_{\Sigma}(\Sigma)\simeq\Sigma and so a horizontal A\mathbf{A}-form on Σ\Sigma we call just a an A\mathbf{A}-form.

Proposition

The horizontal differential of def. 7 commutes with pullback of horizontal differential forms α\alpha along the jet prolongation, def. 6, of any field section σΓ X(E)\sigma \in \Gamma_X(E).

In detail: for

  • d:ιEt ΣΣ *AιEt ΣΣ *Ad \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}' a morphism,

  • α:ιEιEt ΣΣ *A\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A} a horizontal A\mathbf{A}-form on Jet(E)Jet(E), def. 7;

  • σΓ Σ(E)\sigma \in \Gamma_\Sigma(E) a field section,

then there is a natural equivalence

j(σ) *(dα)d(j(σ) *α). j(\sigma)^\ast (d \alpha) \simeq d (j(\sigma)^\ast \alpha) \,.
Proof

Since all objects are in the direct image free:HPDE(H) Σfree\colon \mathbf{H} \to PDE(\mathbf{H})_\Sigma, this is an equivalence of morphisms in the co-Kleisli category of the jet comonad, hence is equivalently an equivalence of co-Kleisli composites of morphisms in H\mathbf{H}.

As such, the left hand side of the equality is given in H\mathbf{H} by the composite morphism

ΣJet(Σ)Jet(σ)Jet(E)Jet(Jet(E))Jet(α˜)Jet(A)d˜A, \Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet(E) \stackrel{}{\to} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,,

thought of as bracketed to the right. By naturality of the Jet-counit this is equivalently

Jet(Σ)Jet(Jet(Σ))Jet(Jet(σ))Jet(Jet(E))Jet(α˜)Jet(A)d˜A, Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet(Jet(\sigma))}{\longrightarrow} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,,

By functorality of Jet()Jet(-) this is equivalent to

Jet(Σ)Jet(Jet(Σ))Jet(α˜Jet(σ))Jet(A)d˜A Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet ( \tilde \alpha \circ Jet(\sigma) )}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\to} \mathbf{A}'

which is the right hand side of the equivalence to be proven.

References

The variational bicomplex was introduced independently in

  • W.M. Tulczyjew, The Euler-Lagrange resolution , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).

  • Alexandre Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints , Sov. Math. Dokl. 19 (1978) 144–148.

  • Alexandre Vinogradov, The CC-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 (1984) 1–129.

Also

  • Floris Takens, A global version of the inverse problem of the calculus of variations J. Diff. Geom. 14 (1979) 543-562

An introduction is in

  • Ian Anderson, Introduction to the variational bicomplex in Mathematical aspects of classical field theory, Contemp. Math. 132 (1992) 51–73, gBooks

Textbook accounts and surveys include

An early discussion with application to covariant phase spaces and their presymplectic structure is in

  • G. J. Zuckerman, Action principles and global geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf)

An invariant version (under group action) is in

  • Irina A. Kogan, Peter J. Olver, The invariant variational bicomplex, pdf

See also

  • Victor Kac, An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology, talk at Algebraic Lie Theory, Newton Institute 2009, video

An application to multisymplectic geometry is discussed in

  • Thomas Bridges, Peter Hydon, Jeffrey Lawson, Multisymplectic structures and the variational bicomplex (pdf)
Revised on August 29, 2015 08:00:50 by Urs Schreiber (88.128.80.56)