Variational bicomplex
Encyclopedia
In mathematics, the 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 exterior forms
on jet manifolds
of sections of a fiber bundle. Lagrangian
s 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 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 theory.
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
on fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. For instance, this is the case of classical field theory
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
on fiber bundles (covariant classical field theory
Covariant classical field theory
In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundles and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the...
).
The variational bicomplex is a cochain complex
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of the differential graded algebra
Differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.- Definition :...
of exterior forms
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
on jet manifolds
Jet bundle
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
of sections of a fiber bundle. Lagrangian
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
s and Euler–Lagrange operators
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
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
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
.
Extended to Lagrangian theory of even and odd fields on graded manifold
Graded manifold
Graded manifolds are extensions of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra. Graded manifolds are not supermanifolds though there is a certain correspondence between the graded manifolds and the DeWitt supermanifolds. Both graded manifolds and...
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.
External links
- Dragon, N., BRS symmetry and cohomology, arXiv: hep-th/9602163
- Sardanashvily, G.Gennadi SardanashvilyGennadi Sardanashvily is a theoretical physicist, a principal research scientist of Moscow State University.- Biography :...
, Graded infinite-order jet manifolds, Int. G. Geom. Methods Mod. Phys. 4 (2007) 1335; arXiv: 0708.2434v1