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Mathematical formulation of Lagrangian mechanics
globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical
Variational_bicomplex
Differential variety
"On variation bicomplexes associated to differential equations". Osaka Journal of Mathematics. 19 (2): 311–363. ISSN 0030-6126. "variational bicomplex in
Diffiety
Differential calculus on function spaces
{\displaystyle y={\hat {y}}.} First variation Isoperimetric inequality Variational principle Variational bicomplex Fermat's principle Principle of least
Calculus_of_variations
Pair in mathematics
second theorem are corollaries of this variational formula. Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian
Lagrangian_system
Russian physicist
and their dynamics is described in terms of jet manifolds and the variational bicomplex (covariant classical field theory) covariant (polysymplectic) Hamiltonian
Gennadi_Sardanashvily
Classical field theories on fiber bundles
Nowadays, it is well known that[citation needed] jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant
Covariant classical field theory
Covariant_classical_field_theory
theory. Noether's second theorem Emmy Noether Lagrangian system Variational bicomplex Gauge symmetry (mathematics) Gomis, J., Paris, J., Samuel, S., Antibracket
Noether_identities
Polish physicist and mathematician (1931–2022)
Vinogradov, Paul Dedecker. Ian Anderson attributed the discovery of the variational bicomplex to both Tulczyjew and Vinogradov independently. In 1974, Tulczyjew
Włodzimierz_Marek_Tulczyjew
Construction in differential topology
Jet (mathematics) Lagrangian system Variational bicomplex Krupka, Demeter (2015). Introduction to Global Variational Geometry. Atlantis Press. ISBN 978-94-6239-073-7
Jet_bundle
Manifold with supersymmetry structure
in terms of graded manifolds. Extended to graded manifolds, the variational bicomplex provides the strict mathematical formulation of Lagrangian classical
Graded_manifold
Russian-Italian mathematician (1938–2019)
equation, i.e., for the space of infinite jets) is the so-called variational bicomplex. Furthermore, Vinogradov introduced a new bracket on the graded
Alexandre Mikhailovich Vinogradov
Alexandre_Mikhailovich_Vinogradov
Repeating pattern of swirling vortices
dynamic equations such as the Ginzburg–Landau equation, or by use of a bicomplex variable. A vortex street forms only at a certain range of flow velocities
Kármán_vortex_street
Quaternions with complex number coefficients
biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See § As a composition algebra below. Note that the matrix product
Biquaternion
Non-coding RNA involved in alternative splicing
the splicing process. Since both U11 and U12 snRNAs come together as a bicomplex, they form a molecular bridge between two ends of introns in the pre-spliceosomal
U11_spliceosomal_RNA
Four-dimensional number system
Theory of Relativity via Internet Archive Cornelius Lanczos (1949) The Variational Principles of Mechanics, University of Toronto Press ISBN 0-8020-1743-6
Quaternion
Algebra based on a vector space with a quadratic form
cases one finds that Cl0(C) ≅ C, the complex numbers Cl1(C) ≅ C ⊕ C, the bicomplex numbers Cl2(C) ≅ M2(C), the biquaternions where Mn(C) denotes the algebra
Clifford_algebra
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