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Topic: Symplectic form

	Symplectic manifold - Wikipedia, the free encyclopedia
		In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form.
		Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g.
		A symplectic manifold endowed with a metric that is compatible with the symplectic form is a Kähler manifold.
	en.wikipedia.org /wiki/Symplectic_manifold (656 words)

	Symplectic topology - Wikipedia
		Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds.
		A symplectic manifold is a pair (M,ω) of a smooth manifold M together with a closed non degenerate differential 2-form ω, the symplectic form.
		Fundamental examples of symplectic manifolds are given by the cotangent bundles of manifolds; these arise in classical mechanics, where the set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
	wikipedia.findthelinks.com /sy/Symplectic_topology.html (503 words)

	Symplectic topology - Wikipedia, the free encyclopedia
		Symplectic topology (also called symplectic geometry although the terms are not completely synonymous) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
		Symplectic topology has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
		Symplectic topology has a number of similarities and differences with Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors).
	en.wikipedia.org /wiki/Symplectic_geometry (338 words)

	PlanetMath: symplectic manifold
		Symplectic manifolds constitute the mathematical structure for modern Hamiltonian mechanics.
		Symplectic manifolds can also be seen as even dimensional analogues to contact manifolds.
		This is version 8 of symplectic manifold, born on 2002-12-05, modified 2006-07-09.
	www.planetmath.org /encyclopedia/SymplecticManifold.html (154 words)

	Symplectic form
		In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle of a manifold Q.
		The tautological one-form is sometimes also called the Liouville one-form, the canonical one-form, or the symplectic potential.
		Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.
	www.algebra.com /algebra/about/history/Symplectic-form.wikipedia (363 words)

	Symplectic manifold - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-01)
		A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form.
		Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorphism between the tangent bundle and the cotangent bundle, thus establishing a one-to-one correspondence between tangent vectors and one-forms.
		A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves; this result has led to the development of a fairly large subdiscipline of symplectic topology.
	encyclopedia.learnthis.info /s/sy/symplectic_manifold.html (747 words)

	Symplectic manifold
		Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds (also called symplectic space).
		Well into the 1970's, symplectic experts were unsure of whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed; in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group of some symplectic 4-manifold, in marked contrast with the Kähler case.
		A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves which has led to the development of a fairly large subdiscipline of symplectic topology.
	www.teachtime.com /en/wikipedia/s/sy/symplectic_manifold.html (645 words)

	PlanetMath: Darboux's Theorem (symplectic geometry)
		Darboux's theorem implies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry, where there is curvature.
		Cross-references: curvature, geometry, invariants, implies, theorem, symplectomorphism, symplectic form, pullback, canonical, coordinate chart, neighborhood, symplectic manifold
		This is version 2 of Darboux's Theorem (symplectic geometry), born on 2002-12-12, modified 2005-05-09.
	www.planetmath.org /encyclopedia/DarbouxCoordinates.html (77 words)

	Symplectic manifold (Site not responding. Last check: 2007-11-01)
		In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form.
		It follows that every symplectic manifold is canonically oriented and comes with a canonical measure, the Liouville measure.
		We have shown that there is a one-to-one correspondence between infinitesimal symplectomorphisms and closed one-forms on a symplectic manifold.
	www.wikimoz.org /wiki/en/wikipedia/s/sy/symplectic_manifold.html (893 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		For example, if we have the cotangent bundle, it is easy to define a canonical symplectic form on it, as an exterior derivative of a one-form.
		The one form assigns to a vector in the tangent bundle to the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold).
		, and the differential is the canonical symplectic form, the sum of dy
	www.informationgenius.com /encyclopedia/c/co/cotangent_space.html (276 words)

	cotangent bundle
		Smooth sections of the cotangent bundle are differential one-forms.
		The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of a one-form.
		The one-form assigns to a vector in the tangent bundle of the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold).
	www.abacci.com /wikipedia/topic.aspx?cur_title=cotangent_bundle (287 words)

	Darboux's theorem - the free encyclopedia (Site not responding. Last check: 2007-11-01)
		Darboux's theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic.
		Let M be a 2n-dimensional symplectic manifold with symplectic form ?.
		This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares.
	www.encyclopedia-of-knowledge.com /default.asp?t=Darboux%27s_theorem (238 words)

	Online Encyclopedia and Dictionary - Symplectic topology
		Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms.
		This is a conesquence of Darboux's theorem which states that every pair of symplectic manifolds are locally isomorphic.
		Results arising from Gromov's theory include Gromov's nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders, and also a proof of a conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows.
	www.fact-archive.com /encyclopedia/Symplectic_geometry (393 words)

	Symplectic vector field - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-01)
		In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian.
		The integral curves of the symplectic vector field are solutions to the Hamilton-Jacobi equations of motion.
		The vector field, taken together with the symplectic manifold and the symplectic form on the manifold, comprise a Hamiltonian system.
	www.gogog.com /project/wikipedia/index.php/Hamiltonian_vector_field (361 words)

	Symplectic topology - the free encyclopedia (Site not responding. Last check: 2007-11-01)
		Darboux's theorem which states that every pair of symplectic manifolds are locally isomorphic.
		Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a
		Gromov-Witten invariants, by which two different symplectic manifolds could in principle be distinguished.
	www.world-knowledge-encyclopedia.com /default.asp?t=Symplectic_topology (341 words)

	Augustin Banyaga - Mathematician of the African Diaspora
		Torre, Carlos A.; Banyaga, Augustin A symplectic structure on coadjoint orbits of diffeomorphism subgroups.
		Banyaga, Augustin; Rukimbira, Philippe On characteristics of circle invariant presymplectic forms.
		Banyaga, Augustin On characteristics of hypersurfaces in symplectic manifolds.
	www.math.buffalo.edu /mad/PEEPS/banyaga_augustin.html (1021 words)

	math lessons - Symplectic vector space
		A symplectic manifold is a smooth manifold with a smoothly-varying symplectic form on each tangent space
		For each n, the symplectic group of degree 2n is the group of symplectic 2n by 2n matrices.
		A symplectic representation is a group representation where each group element acts as a symplectic transformation.
	www.mathdaily.com /lessons/Symplectic_vector_space (560 words)

	Symplectic space (Site not responding. Last check: 2007-11-01)
		A symplectic space is a 2n-dimensional manifold with a nondegenerate 2-form, ω, called the symplectic form.
		with and for all i,j=1,...,n-1 where δ is the Kronecker delta function is a linear symplectic space.
		One of the primary applications of symplectic spaces is in Hamiltonian mechanics.
	www.icyclopedia.com /encyclopedia/s/sy/symplectic_space.html (65 words)

	Tautological one-form
		In Mathematics, the tautological one-form is a special 1-form defined on symplectic manifolds that plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics.
		In Local coordinates, the canonical symplectic form is exact; the tautological one-form is the one-form whose differential is (minus) the symplectic form on the symplectic manifold.
		The tautological one-form is sometimes also called the canonical one-form or the symplectic potential.
	www.ufaqs.com /wiki/en/ta/Tautological%20oneform.htm (249 words)

	GAP Manual: 67.18 ClassicalForms
		(that is, an invariant symplectic or unitary bilinear form or an invariant symmetric bilinear form together with an invariant quadratic form, invariant modulo scalars in each case) or try to prove that no such form exists.
		A quadratic form is returned as upper triangular matrix Q such that g * Q * g^{tr} equals Q modulo scalars after g * Q * g^{tr} has been normalized into an upper triangular matrix.
		The "symplectic" indicates that an invariant symplectic form exists, the "unknown" indicates that an invariant "unitary" form might exist.
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C067S018.htm (499 words)

	SymplecticManifold (Site not responding. Last check: 2007-11-01)
		Symplectic manifolds can also be seen as even dimensional analogues to
		Definition 1 A symplectic manifold is a pair
		This is easy to understand in view of the physics.
	www.objectsspace.com /encyclopedia/mathematics/entries/53/SymplecticManifold/SymplecticManifold.html (73 words)

	[No title]
		Introduction Throughout the paper the term "symplectic manifold" means a closed symplectic manifold (M, !) such that the cohomology class [!] 2 H2(M; R) lies in the integral lattice H2(M)=tors.
		By the definition, a symplectically aspherical manifold is a symplectic manifold whose symplectic form is symplectically aspherical.
		One more sourse of symplectically aspherical groups comes from the observation of Gompf [G2, Lemma 1] who proved that a branched cov- ering over a 4-dimensional symplectically aspherical manifold is sym- plectically aspherical.
	hopf.math.purdue.edu /Ibanez-Rudyak-Tralle/aspherical.txt (3988 words)

	Växjö universitet: Disputation vid Växjö universitet
		Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with.
		A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties.
		Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes.
	www.vxu.se /avhandlingar/serge_degosson.xml (296 words)

	Springer Online Reference Works
		The other real forms of this group are also sometimes called symplectic groups.
		Hamilton equations) the phase space is a symplectic manifold, a manifold
		As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.
	eom.springer.de /s/s091820.htm (301 words)

	Reference.com/Encyclopedia/Cotangent bundle
		Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.
		Intrinsically, the canonical one-form is given as a pullback.
		The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of the canonical one-form, the symplectic potential.
	www.reference.com /browse/wiki/Cotangent_bundle (830 words)

	Structure of Dynamical Systems (Progress in Mathematics) by J.M. Souriau [ISBN: 0817636951] - Find Cheap Textbook ...
		The aim of the book is to treat all three basic theories of physics, namely, classical mechanics, statistical mechanics, and quantum mechanics from the same perspective, that of symplectic geometry, thus showing the unifying power of the symplectic geometric approach.
		In Chapter 3 a coherent symplectic description of Galilean and relativistic mechanics is given, culminating in the classification of elementary particles (relativistic and non-relativistic, with or without spin, with or without mass).
		In Chapter 4 statistical mechanics is put into symplectic form, finishing with a symplectic description of the kinetic theory of gases and the computation of specific heats.
	www.gettextbooks.com /isbn_0817636951.html (369 words)

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