variational bicomplex


Variational calculus


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For XX a (spacetime) manifold and EXE \to X a bundle (in physics called the field bundle) with jet bundle j EXj_\infty E \to X, the variational bicomplex is essentially the de Rham complex (Ω (j E),d)(\Omega^\bullet(j_\infty E),\mathbf{d}) of j Ej_\infty E with differential forms Ω n(j E)= h+v=nΩ h,v(E)\Omega^n(j_\infty 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 j Ej_\infty E, dd is called the horizontal differential and δ\delta is called the vertical differential.

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 a vertical 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.


Let XX be a smooth manifold and p:EXp : E \to X some smooth bundle over XX. Write j EXj_\infty E \to X for the corresponding jet bundle.

The spaces of sections Γ(E)\Gamma(E) and Γ(j E)\Gamma(j_\infty 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) &\to& * \\ \downarrow && \downarrow^{id} \\ [X,E] &\stackrel{p_*}{\to}& [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(j E) j_\infty : \Gamma(E) \to \Gamma(j_\infty E)

it yields a smooth map (homomorphism of diffeological spaces)

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


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

d:=d+δ, \mathbf{d} {{:}=} 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.


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

We write

Ω loc ,:=im(e *) \Omega^{\bullet, \bullet}_{loc} := 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) to (Takens). The above formulation as the image of the evident bicomplex of forms on X×Γ(E)X \times \Gamma(E) is due to (Zuckerman, p. 5).


The fundamental variational formula


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.


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 \,.


  • 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, theorem 3).

Here EE is the Euler-Lagrange operator .



Ω=δΘ. \Omega = \delta \Theta \,.

By prop. 1 have

dΩ=δE(L). d \Omega = -\delta E(L) \,.

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, lemma 8]).

Presymplectic covariant phase space


The form Ω\Omega is a conserved current.


By remark 1 and prop. 2.


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.


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 \,.


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


A vertical vector fields vT vET vj Ev \in T_v E \subset T_v j_\infty E is a symmetry if

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

The presymplectic form ω\omega from def. 4 is degenerate on symmetries.

This appears as (Zuckerman, theorem 13).


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).

  • A.M. 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.

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


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

An textbook account is in section 5.4 of

  • Peter Olver, Applications of Lie groups to differential equations, Springer Graduate Texts in Mathematics 107 (1986)

An invariant version (under group action) is in

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

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 introduction is in

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

  • Ian Anderson, The variational bicomplex, Utah State University (1989) (pdf) (textbook account)

  • 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 March 10, 2014 10:23:12 by Fosco Loregian (