View source for Variational bicomplex

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In mathematics, the [[Lagrangian system|Lagrangian theory]] on [[fiber bundle]]s is globally formulated in algebraic terms of the '''variational bicomplex''', without appealing to the [[calculus of variations]]. For instance, this is the case of  [[classical field theory]] on fiber bundles ([[covariant classical field theory]]).

The variational bicomplex is a [[cochain complex]] of the [[differential graded algebra]] of [[differential form|exterior forms]] on [[jet bundle|jet manifolds]] of sections of a fiber bundle. [[Lagrangian system|Lagrangian]]s and [[Lagrangian system|Euler–Lagrange operators]] on a fiber bundle are defined as elements of this bicomplex. [[Cohomology]] of the variational bicomplex leads to the global first variational formula and first [[Noether's theorem]].

Extended to Lagrangian theory of even and odd fields on [[graded manifold]]s, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian [[BRST formalism|BRST theory]].

== See also ==

*[[Calculus of variations]]
*[[Lagrangian system]]
*[[Jet bundle]]

== References ==

* {{Citation | last1=Takens | first1=Floris | author1-link=Floris Takens | title=A global version of the inverse problem of the calculus of variations | mr=600611 | year=1979 | journal=Journal of Differential Geometry | issn=0022-040X | volume=14 | issue=4 | pages=543–562 | doi=10.4310/jdg/1214435235 | s2cid=118169017 | doi-access=free }}
* Anderson, I., "Introduction to variational bicomplex", ''Contemp. Math''. '''132''' (1992) 51.
* Barnich, G., Brandt, F., Henneaux, M., "Local BRST cohomology", ''Phys. Rep''. '''338''' (2000) 439.
* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Advanced Classical Field Theory'', World Scientific, 2009, {{isbn|978-981-283-895-7}}.

== External links ==
* Dragon, N., BRS symmetry and cohomology, {{arXiv|hep-th/9602163}} 
* [[Gennadi Sardanashvily|Sardanashvily, G.]], Graded infinite-order jet manifolds, Int. G. Geom. Methods Mod. Phys. '''4''' (2007) 1335; {{arXiv|0708.2434}}

[[Category:Calculus of variations]]
[[Category:Differential geometry]]


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