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• Symplectic_manifold abstract "In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to Hamilton's equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.".
• Symplectic_manifold wikiPageExternalLink 9907034.
• Symplectic_manifold wikiPageExternalLink mcduff.pdf.
• Symplectic_manifold wikiPageExternalLink 0908.1886.
• Symplectic_manifold wikiPageID "28356".
• Symplectic_manifold wikiPageLength "14409".
• Symplectic_manifold wikiPageOutDegree "94".
• Symplectic_manifold wikiPageRevisionID "704323281".
• Symplectic_manifold wikiPageWikiLink Almost_complex_manifold.
• Symplectic_manifold wikiPageWikiLink Almost_symplectic_manifold.
• Symplectic_manifold wikiPageWikiLink Analytical_mechanics.
• Symplectic_manifold wikiPageWikiLink Betti_number.
• Symplectic_manifold wikiPageWikiLink Block_matrix.
• Symplectic_manifold wikiPageWikiLink Calabi–Yau_manifold.
• Symplectic_manifold wikiPageWikiLink Category:Differential_topology.
• Symplectic_manifold wikiPageWikiLink Category:Hamiltonian_mechanics.
• Symplectic_manifold wikiPageWikiLink Category:Smooth_manifolds.
• Symplectic_manifold wikiPageWikiLink Category:Symplectic_geometry.
• Symplectic_manifold wikiPageWikiLink Caustic_(mathematics).
• Symplectic_manifold wikiPageWikiLink Classical_mechanics.
• Symplectic_manifold wikiPageWikiLink Closed_and_exact_differential_forms.
• Symplectic_manifold wikiPageWikiLink Commutative_diagram.
• Symplectic_manifold wikiPageWikiLink Contact_geometry.
• Symplectic_manifold wikiPageWikiLink Cotangent_bundle.
• Symplectic_manifold wikiPageWikiLink Covariant_Hamiltonian_field_theory.
• Symplectic_manifold wikiPageWikiLink Critical_value.
• Symplectic_manifold wikiPageWikiLink Darbouxs_theorem.
• Symplectic_manifold wikiPageWikiLink Degenerate_bilinear_form.
• Symplectic_manifold wikiPageWikiLink Diffeomorphism.
• Symplectic_manifold wikiPageWikiLink Differentiable_function.
• Symplectic_manifold wikiPageWikiLink Differentiable_manifold.
• Symplectic_manifold wikiPageWikiLink Differential_equation.
• Symplectic_manifold wikiPageWikiLink Differential_form.
• Symplectic_manifold wikiPageWikiLink Dusa_McDuff.
• Symplectic_manifold wikiPageWikiLink Euler_characteristic.
• Symplectic_manifold wikiPageWikiLink Exterior_algebra.
• Symplectic_manifold wikiPageWikiLink Exterior_derivative.
• Symplectic_manifold wikiPageWikiLink Fedosov_manifold.
• Symplectic_manifold wikiPageWikiLink Fiber_bundle.
• Symplectic_manifold wikiPageWikiLink Fibration.
• Symplectic_manifold wikiPageWikiLink File:TIKZ_PICT_FBN.png.
• Symplectic_manifold wikiPageWikiLink Gennadi_Sardanashvily.
• Symplectic_manifold wikiPageWikiLink Hamiltonian_mechanics.
• Symplectic_manifold wikiPageWikiLink Hamiltonian_vector_field.
• Symplectic_manifold wikiPageWikiLink Hermitian_manifold.
• Symplectic_manifold wikiPageWikiLink Hyperkähler_manifold.
• Symplectic_manifold wikiPageWikiLink Identity_matrix.
• Symplectic_manifold wikiPageWikiLink Immersion_(mathematics).
• Symplectic_manifold wikiPageWikiLink Integrability_conditions_for_differential_systems.
• Symplectic_manifold wikiPageWikiLink Integral_curve.
• Symplectic_manifold wikiPageWikiLink Interior_product.
• Symplectic_manifold wikiPageWikiLink Jerrold_E._Marsden.
• Symplectic_manifold wikiPageWikiLink Kähler_manifold.
• Symplectic_manifold wikiPageWikiLink Lie_derivative.
• Symplectic_manifold wikiPageWikiLink Liouvilles_theorem_(Hamiltonian).
• Symplectic_manifold wikiPageWikiLink Mathematical_optimization.
• Symplectic_manifold wikiPageWikiLink Mathematics.
• Symplectic_manifold wikiPageWikiLink Maurice_A._de_Gosson.
• Symplectic_manifold wikiPageWikiLink Metric_tensor.
• Symplectic_manifold wikiPageWikiLink Mirror_symmetry_(string_theory).
• Symplectic_manifold wikiPageWikiLink Newtons_laws_of_motion.
• Symplectic_manifold wikiPageWikiLink Phase_space.
• Symplectic_manifold wikiPageWikiLink Poisson_bracket.
• Symplectic_manifold wikiPageWikiLink Poisson_manifold.
• Symplectic_manifold wikiPageWikiLink Pullback_(differential_geometry).
• Symplectic_manifold wikiPageWikiLink Quadratic_form.
• Symplectic_manifold wikiPageWikiLink Ralph_Abraham.
• Symplectic_manifold wikiPageWikiLink SYZ_conjecture.
• Symplectic_manifold wikiPageWikiLink Section_(fiber_bundle).
• Symplectic_manifold wikiPageWikiLink Skew-symmetric_matrix.
• Symplectic_manifold wikiPageWikiLink Submanifold.
• Symplectic_manifold wikiPageWikiLink Symplectic_geometry.
• Symplectic_manifold wikiPageWikiLink Symplectic_group.
• Symplectic_manifold wikiPageWikiLink Symplectic_matrix.
• Symplectic_manifold wikiPageWikiLink Symplectic_vector_space.
• Symplectic_manifold wikiPageWikiLink Symplectomorphism.
• Symplectic_manifold wikiPageWikiLink Tautological_one-form.
• Symplectic_manifold wikiPageWikiLink Two-form.
• Symplectic_manifold wikiPageWikiLink Vector_field.
• Symplectic_manifold wikiPageWikiLink Volume_form.
• Symplectic_manifold wikiPageWikiLink Wirtinger_inequality_(2-forms).
• Symplectic_manifold wikiPageWikiLinkText "Lagrangian fibration".
• Symplectic_manifold wikiPageWikiLinkText "Lagrangian immersion".
• Symplectic_manifold wikiPageWikiLinkText "Lagrangian submanifold".
• Symplectic_manifold wikiPageWikiLinkText "Lagrangian submanifolds".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic manifold".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic manifold#Lagrangian and other submanifolds".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic manifold#Special cases and generalizations".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic_manifold#Lagrangian_fibration".
• Symplectic_manifold wikiPageWikiLinkText "Symplectic_manifold#Lagrangian_mapping".
• Symplectic_manifold wikiPageWikiLinkText "multisymplectic".
• Symplectic_manifold wikiPageWikiLinkText "polysymplectic".
• Symplectic_manifold wikiPageWikiLinkText "special Lagrangian geometry".
• Symplectic_manifold wikiPageWikiLinkText "symplectic form".
• Symplectic_manifold wikiPageWikiLinkText "symplectic manifold".
• Symplectic_manifold wikiPageWikiLinkText "symplectic manifold#Special Lagrangian submanifolds".
• Symplectic_manifold wikiPageWikiLinkText "symplectic space".
• Symplectic_manifold wikiPageWikiLinkText "symplectic structure".
• Symplectic_manifold wikiPageWikiLinkText "symplectic".

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