nLab variational bicomplex

Context

Variational calculus

variational calculus

Contents

Idea

For $X$ a (spacetime) manifold and $E \to X$ a bundle (in physics called the field bundle) with jet bundle $j_\infty E \to X$, the variarional bicomplex is essentially the de Rham complex $(\Omega^\bullet(j_\infty E),\mathbf{d})$ of $j_\infty E$ with differential forms $\Omega^n(j_\infty E) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ bigraded by horizontal degree $h$ (with respect to $X$) and vertical degree $v$ (along the fibers of $j_\infty E$)). Accordingly the differential decomposes as

$\mathbf{d} = d + \delta \,,$

where $\mathbf{d}$ is the de Rham differential on $j_\infty E$, $d$ is called the horizontal differential and $\delta$ is called the vertical differential.

Much of classical mechanics and classical field theory on $X$ is formalized in terms of the variational bicomplex. For instance

• a field configuration is a section of $E$;

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

• a local action functional is a map

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

of the form

$S(\phi) = \int_X L(j_\infty \phi) \,,$
• $E(L) := \delta L \mod im d = 0$
• the covariant phase space is the locus

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

$d \eta = 0 \mod E(L)$
• a symmetry is a vertical vector field $v$ such that

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

Definition

Let $X$ be a smooth manifold and $p : E \to X$ some smooth bundle over $X$. Write $j_\infty E \to X$ for the corresponding jet bundle.

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

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

This induces the evaluation map

$X \times \Gamma(E) \to E \,.$

and composed with the jet prolongation

$j_\infty : \Gamma(E) \to \Gamma(j_\infty E)$

it yields a smooth map (homomorphism of diffeological spaces)

(1)$e_\infty : X \times \Gamma(E) \stackrel{(id,j_\infty)}{\to} X \times \Gamma(j_\infty E) \stackrel{ev}{\to} j_\infty E \,.$

Write

$\Omega^{\bullet, \bullet}(X \times \Gamma(E))$

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

$\mathbf{d} {{:}=} d + \delta \,,$

where the $\mathbf{d}$, $d$ and $\delta$, are the de Rham differentials of $X\times\Gamm(E)$, $X$ and $\Gamma(E)$, respectively.

Definition

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

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

We write

$\Omega^{\bullet, \bullet}_{loc} := im (e_\infty^*)$

and speak of the bicomplex of local forms on sections on $E$.

The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in (Olver) to (Takens). The above formulation as the image of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to (Zuckerman, p. 5).

Properties

The fundamental variational formula

Definition

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

$\alpha_\phi(\delta \phi)$

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

Proposition

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

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

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

Moreover

• $E(L)$ is independent of changes of $L$ by $d$-exact terms:

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

This is (Zuckerman, theorem 3).

Here $E$ is the Euler-Lagrange operator .

Definition

Write

$\Omega = \delta \Theta \,.$
Remark

By prop. 1 have

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

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

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

This is ([Zuckerman, lemma 8]).

Presymplectic covariant phase space

Corollary

The form $\Omega$ is a conserved current.

Proof

By remark 1 and prop. 2.

Definition

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

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

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

Proposition

The 2-form $\omega$ is indeed closed

$\delta \omega = 0$

and in fact exact:

$\theta := \int_\Sigma \Theta$

is its presymplectic potential .

$\delta \theta = \omega \,.$

Symmetries

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

Definition

A vertical vector fields $v \in T_v E \subset T_v j_\infty E$ is a symmetry if

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

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

This appears as (Zuckerman, theorem 13).

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).

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

Also

• 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 September 19, 2013 03:18:16 by Igor Khavkine (82.157.45.36)