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Symplectic Space (other)
A symplectic space may refer to: * Symplectic manifold * Symplectic vector space In mathematics, a symplectic vector space is a vector space V over a Field (mathematics), field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping \omega : ... {{Disambig Differential geometry Differential topology Symplectic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , 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. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space V over a Field (mathematics), field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping \omega : V \times V \to F that is ; bilinear form, Bilinear: linear map, Linear in each argument separately; ; alternating form, Alternating: \omega(v, v) = 0 holds for all v \in V; and ; Nondegenerate form, Non-degenerate: \omega(v, u) = 0 for all v \in V implies that u = 0. If the underlying field (mathematics), field has characteristic (algebra), characteristic not 2, alternation is equivalent to skew symmetry, skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric bilinear form, symmetric form, but not vice versa. Working in a fixed basis (linear algebra), basis, \omega can be represented by a matrix (mathematics), matrix. The conditions abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected) manifolds in each dimension separately: * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
