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65 relations: Almost complex manifold, Almost symplectic manifold, Analytical mechanics, Betti number, Block matrix, Calabi–Yau manifold, Caustic (mathematics), Classical mechanics, Closed and exact differential forms, Commutative diagram, Contact geometry, Cotangent bundle, Critical value, Darboux's theorem, Degenerate bilinear form, Diffeomorphism, Differentiable function, Differential equation, Differential form, Dusa McDuff, Eckhard Meinrenken, Euler characteristic, Exterior algebra, Exterior derivative, Fedosov manifold, Fibration, Gennadi Sardanashvily, Hamiltonian field theory, Hamiltonian mechanics, Hamiltonian vector field, Hermitian manifold, Hyperkähler manifold, Identity matrix, Immersion (mathematics), Integrability conditions for differential systems, Integral curve, Interior product, Jerrold E. Marsden, Kähler manifold, Lie derivative, Liouville's theorem (Hamiltonian), Mathematical optimization, Mathematics, Maurice A. de Gosson, Metric tensor, Mirror symmetry (string theory), Nondegenerate form, Phase space, Poisson bracket, Poisson manifold, ..., Quadratic form, Ralph Abraham (mathematician), Section (fiber bundle), Skew-symmetric matrix, Submanifold, Symplectic geometry, Symplectic group, Symplectic matrix, Symplectic vector space, Symplectomorphism, SYZ conjecture, Tautological one-form, Vector field, Volume form, Wirtinger inequality (2-forms). Expand index (15 more) »
Almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space.
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Almost symplectic manifold
In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M that is everywhere non-singular.
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Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics.
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Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
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Block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
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Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics.
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Caustic (mathematics)
In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold.
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Classical mechanics
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
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Closed and exact differential forms
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα.
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Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
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Contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'.
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Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.
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Critical value
Critical value may refer to.
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Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem.
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Degenerate bilinear form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V) given by is not an isomorphism.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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Differentiable function
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
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Differential equation
A differential equation is a mathematical equation that relates some function with its derivatives.
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Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
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Dusa McDuff
Dusa McDuff FRS CorrFRSE (born 18 October 1945) is an English mathematician who works on symplectic geometry.
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Eckhard Meinrenken
Eckhard Meinrenken is a Canadian mathematician specializing in Symplectic Geometry, Lie Theory, Mathematical Physics.
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Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
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Fedosov manifold
In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (i.e., ω is a symplectic form, a non-degenerate closed exterior 2-form, on a C∞-manifold M), and ∇ is a symplectic torsion-free connection on M. (A connection ∇ is called compatible or symplectic if X ⋅ ω(Y,Z).
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Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle.
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Gennadi Sardanashvily
Gennadi Sardanashvily (Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 - September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University.
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Hamiltonian field theory
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics.
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Hamiltonian mechanics
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
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Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian.
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Hermitian manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold.
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Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(''k'') (here Sp(k) denotes a compact form of a symplectic group, identified with the group of quaternionic-linear unitary endomorphisms of a k-dimensional quaternionic Hermitian space).
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Identity matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.
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Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.
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Integrability conditions for differential systems
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.
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Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
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Interior product
In mathematics, the interior product (aka interior derivative/, interior multiplication, inner multiplication, inner derivative, or inner derivation) is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold.
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Jerrold E. Marsden
Jerrold Eldon Marsden (August 17, 1942 – September 21, 2010) was a mathematician.
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Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.
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Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow defined by another vector field.
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Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.
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Mathematical optimization
In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maurice A. de Gosson
Maurice A. de Gosson (born 13 March 1948), (also known as Maurice Alexis de Gosson de Varennes) is an Austrian mathematician and mathematical physicist, born in 1948 in Berlin.
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Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
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Mirror symmetry (string theory)
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds.
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Nondegenerate form
In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if.
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Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
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Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system.
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Poisson manifold
In geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M, subject to the Leibniz rule Said in another manner, it is a Lie algebra structure on the vector space of smooth functions on M such that X_ \stackrel \: (M) \to (M) is a vector field for each smooth function f, which we call the Hamiltonian vector field associated to f. These vector fields span a completely integrable singular foliation, each of whose maximal integral sub-manifolds inherits a symplectic structure.
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Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.
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Ralph Abraham (mathematician)
Ralph H. Abraham (born July 4, 1936) is an American mathematician.
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Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.
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Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.
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Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.
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Symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
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Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.
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Symplectic matrix
In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition where MT denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skew-symmetric matrix.
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Symplectic vector space
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form.
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Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.
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SYZ conjecture
The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics.
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Tautological one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T*Q the structure of a symplectic manifold.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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Volume form
In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).
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Wirtinger inequality (2-forms)
In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold M, the exterior kth power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable) (2k)-vector ζ of unit volume, is bounded above by k!.
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Homogeneous symplectic manifold, Isotropic submanifold, Lagrangian fibration, Lagrangian map, Lagrangian mapping, Lagrangian submanifold, Liouville measure, Multisymplectic manifold, Polysymplectic manifold, Special Lagrangian submanifold, Symplectic submanifold.
References
[1] https://en.wikipedia.org/wiki/Symplectic_manifold
