For a (spacetime) manifold and a bundle (in physics called the field bundle) with jet bundle , the variational bicomplex is essentially the de Rham complex of with differential forms bigraded by horizontal degree (with respect to ) and vertical degree (along the fibers of )). Accordingly the differential decomposes as
where is the de Rham differential on , is called the horizontal differential and is called the vertical differential.
Much of classical mechanics and classical field theory on is formalized in terms of the variational bicomplex. For instance
Let be a smooth manifold and some smooth bundle over . Write for the corresponding jet bundle.
The spaces of sections and canonically inherit a generalized smooth structure that makes them diffeological spaces: we have a pullback diagram of diffeological spaces
This induces the evaluation map
and composed with the jet prolongation
it yields a smooth map (homomorphism of diffeological spaces)
for the cochain complex of smooth differential forms on the product , bigraded with respect to the differentials on the two factors
where the , and , are the de Rham differentials of , and , respectively.
The variational bicomplex of is the sub–bi-complex of that is the image of the pullback of forms along the map (1):
and speak of the bicomplex of local forms on sections on .
The bicomplex structure on is attributed in (Olver) to (Takens). The above formulation as the image of the evident bicomplex of forms on is due to (Zuckerman, p. 5).
The fundamental variational formula
A source form is an element in such that
depends only on the 0-jet of .
Then there is a unique source form such that
This is (Zuckerman, theorem 3).
Here is the Euler-Lagrange operator .
vanishes when restricted to vertical tangent vectors based in covariant phase space (but not necessarily tangential to it).
This is ([Zuckerman, lemma 8]).
Presymplectic covariant phase space
By remark 1 and prop. 2.
The 2-form is indeed closed
and in fact exact:
is its presymplectic potential .
A vertical vector fields is a symmetry if
The presymplectic form 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 -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. 259284. (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)