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- Lagrangian_system abstract "In mathematics, a Lagrangian system is a pair 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 Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle over the time axis (in particular, if a reference frame is fixed). In classical field theory, all field systems are the Lagrangian ones.A Lagrangian density (or, simply, a Lagrangian) of order is defined as an -form, dim, on the -order jet manifold of . A Lagrangian can be introduced as an element of the variational bicomplex of the differential graded algebra of exterior forms on jet manifolds of . The coboundary operator of this bicomplex contains the variational operator which, acting on , defines the associated Euler–Lagrange operator . Given bundle coordinates on a fiber bundle and the adapted coordinates (, ) on jet manifolds , a Lagrangian and its Euler–Lagrange operator read where denote the total derivatives. For instance, a first-order Lagrangian and its second-order Euler–Lagrange operator take the form The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations .Cohomology of the variational bicomplex leads to the so-calledvariational formula where is the total differential and is a Lepage equivalent of . Noether's first theorem and Noether's second theorem are corollaries of this variational formula.Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.In a different way, Lagrangians, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.".
- Lagrangian_system wikiPageExternalLink 0908.1886.
- Lagrangian_system wikiPageExternalLink 1206.2508.
- Lagrangian_system wikiPageID "24585634".
- Lagrangian_system wikiPageRevisionID "600639293".
- Lagrangian_system hasPhotoCollection Lagrangian_system.
- Lagrangian_system subject Category:Calculus_of_variations.
- Lagrangian_system subject Category:Differential_operators.
- Lagrangian_system subject Category:Dynamical_systems.
- Lagrangian_system subject Category:Lagrangian_mechanics.
- Lagrangian_system type Abstraction100002137.
- Lagrangian_system type Attribute100024264.
- Lagrangian_system type DifferentialOperators.
- Lagrangian_system type DynamicalSystem106246361.
- Lagrangian_system type DynamicalSystems.
- Lagrangian_system type Function113783816.
- Lagrangian_system type MathematicalRelation113783581.
- Lagrangian_system type Operator113786413.
- Lagrangian_system type PhaseSpace100029114.
- Lagrangian_system type Relation100031921.
- Lagrangian_system type Space100028651.
- Lagrangian_system comment "In mathematics, a Lagrangian system is a pair 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 Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle over the time axis (in particular, if a reference frame is fixed).".
- Lagrangian_system label "Lagrangian system".
- Lagrangian_system label "Лагранжева система".
- Lagrangian_system sameAs m.08052s6.
- Lagrangian_system sameAs Q7424540.
- Lagrangian_system sameAs Q7424540.
- Lagrangian_system sameAs Lagrangian_system.
- Lagrangian_system wasDerivedFrom Lagrangian_system?oldid=600639293.
- Lagrangian_system isPrimaryTopicOf Lagrangian_system.