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Symplectic manifold

In differential geometry, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate differential 2-form, ω {\displaystyle \omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations…

Examples, Definition & Lagrangian submanifolds

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Overview

Motivation

Definition

Submanifolds

Lagrangian submanifolds

Symmetries

Examples

Generalizations

General and cited references

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Symplectic manifold

Nodes101
Edges100
Triples132
Avg. degree1.98
Density0.019802
Components1

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Symplectic manifold

Top relations

related to General and cited references · 51
Symplectic manifold → Abraham, Advances, Alan Weinstein, Arnold, Auroux, Basel, Benjamin-Cummings, Birkhäuser Verlag, Caustics, Ch, Denis, Dordrecht, Dusa, Eckhard, Eduard, Foundations, Gosson, Hamiltonian Dynamics, Helmut, Hofer
related to Kähler manifolds · 11
Symplectic manifold → Any, CP, For, Fubini, JX, JY, Kähler, Riemannian, Study, They, When
related to Lagrangian mapping · 11
Symplectic manifold → B1, B2, K1, K2, L1, L2, Lagrangian, Let, Symbolically, The, Two Lagrangian
related to Motivation · 11
Symplectic manifold → Applying Cartan's, Finally, Hamilton, Hamiltonian, In, Letting, Lie, Since, So, Symplectic, TM
see also · 10
Symplectic manifold → Almost, Branch, Canonical, Covariant Hamiltonian, Formalism, Hamiltonian, Isomorphism, Mathematical, Mathematics, Operation
related to Generalizations · 8
Symplectic manifold → Any, Dirac, Hamiltonian, Legendre, Poisson, Presymplectic, The, They
related to Basic properties · 6
Symplectic manifold → By Darboux's, Consequently, If, Riemannian, Thus, Unlike
related to Definition · 6
Symplectic manifold → Assigning, Here, Let, Since, That, The
related to Submanifolds · 5
Symplectic manifold → By, It, Lagrangian, Let, There
related to Symmetries · 5
Symplectic manifold → Any Hamiltonian, Given, Hamiltonian, Such, The

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Important terminology

displaystyle symplectic omega manifold form lagrangian submanifold field vector manifolds geometry closed called smooth hamiltonian structure cotangent submanifolds 1-form liouville

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Symplectic manifoldis asmooth manifold0.90text
Symplectic manifoldis apair0.90text
Symplectic manifoldrelated to Basic propertiesIf0.60section
Symplectic manifoldrelated to Basic propertiesThus0.60section
Symplectic manifoldrelated to Basic propertiesUnlike0.60section
Symplectic manifoldrelated to Basic propertiesRiemannian0.60section
Symplectic manifoldrelated to Basic propertiesBy Darboux's0.60section
Symplectic manifoldrelated to Basic propertiesConsequently0.60section
Symplectic manifoldrelated to DefinitionLet0.60section
Symplectic manifoldrelated to DefinitionHere0.60section
Symplectic manifoldrelated to DefinitionThat0.60section
Symplectic manifoldrelated to DefinitionThe0.60section

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