In mathematics, the
Lagrangian theory on
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in 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 p ...
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 functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. For instance, this is the case of
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
on fiber bundles (
covariant classical field theory).
The variational bicomplex is a
cochain complex of the
differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
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Definition
A differential graded alg ...
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 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
*
Calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
*
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
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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