In mathematics, the Lagrangian theory on fiber bundles 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 exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and 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 manifolds , the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory .
See also
References
Takens, Floris (1979), "A global version of the inverse problem of the calculus of variations", Journal of Differential Geometry , 14 (4): 543– 562, doi :10.4310/jdg/1214435235 ISSN 0022-040X , MR 0600611 , S2CID 118169017 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., Sardanashvily, G. , Advanced Classical Field Theory , World Scientific, 2009, ISBN 978-981-283-895-7 .
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