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 In mathematical physics, covariant classical field theory represents classical fields 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 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 included in the kernel of th ...
of the
differential graded algebra of
exterior forms on
jet manifolds of sections of a fiber bundle.
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
s 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 viewe ...
of the variational bicomplex leads to the global first variational formula and first
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
.
Extended to Lagrangian theory of even and odd fields on
graded manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifold, manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaf (mathematics) ...
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.
See also
*
Calculus of variations
*
Lagrangian system
In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of .
In classical mechanics, many dynamical systems are Lagr ...
*
Jet bundle
In differential topology, the jet bundle is a certain construction that 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. ...
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 equations
Differential geometry
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