In mathematics, the Lagrangian theory on

fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...

s is globally formulated in algebraic terms of the variational bicomplex, without appealing to the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

. For instance, this is the case of

classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...

on fiber bundles (

covariant classical field theory In mathematical physics, covariant classical field theory represents classical field theory, classical fields by Section (fiber bundle), sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of field ( ...

). The variational bicomplex is a

cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel ...

of the

differential graded algebra In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geo ...

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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

of the variational bicomplex leads to the global first variational formula and first

Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...

. Extended to Lagrangian theory of even and odd fields on

graded manifold In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutati ...

s, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory.

References

* * 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, .

External links

* Dragon, N., BRS symmetry and cohomology, * Sardanashvily, G., Graded infinite-order jet manifolds, Int. G. Geom. Methods Mod. Phys. 4 (2007) 1335; Calculus of variations Differential geometry {{mathanalysis-stub

See also

References

External links