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Reference.org
Variational bicomplex
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<p>In mathematics, the <a href="/facts/Lagrangian_system/3w7eSwYg">Lagrangian theory</a> on <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundles</a> is globally formulated in algebraic terms of the variational bicomplex, without appealing to the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>. For instance, this is the case of <a href="/facts/Classical_field_theory/7HuKY1qu">classical field theory</a> on fiber bundles (<a href="/facts/Covariant_classical_field_theory/dYegRrqf">covariant classical field theory</a>). </p><p>The variational bicomplex is a <a href="/facts/Cochain_complex/YAlchtCT">cochain complex</a> of the <a href="/facts/Differential_graded_algebra/zySud4Kl">differential graded algebra</a> of <a href="/facts/Differential_form/T9gyHtMv">exterior forms</a> on <a href="/facts/Jet_bundle/7RHyhv07">jet manifolds</a> of sections of a fiber bundle. <a href="/facts/Lagrangian_system/3w7eSwYg">Lagrangians</a> and <a href="/facts/Lagrangian_system/3w7eSwYg">Euler–Lagrange operators</a> on a fiber bundle are defined as elements of this bicomplex. <a href="/facts/Cohomology/J7FJzTNh">Cohomology</a> of the variational bicomplex leads to the global first variational formula and first <a href="/facts/Noether%27s_theorem/VFjqrEvz">Noether's theorem</a>. </p><p>Extended to Lagrangian theory of even and odd fields on <a href="/facts/Graded_manifold/cr99oc5R">graded manifolds</a>, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian <a href="/facts/BRST_formalism/BdNA0oDH">BRST theory</a>. </p>