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Symplectic Vector Bundle
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The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to: * Symplectic category * Symplectic Clifford algebra, see Weyl algebra * Symplectic geometry * Symplectic group, and corresponding symplectic Lie algebra * Symplectic integrator * Symplectic manifold * Symplectic matrix * Symplectic representation * Symplectic vector space, a vector space with a symplectic bilinear form It can also refer to: * Symplectic bone, a bone found in fish skulls * Symplectite, in reference to a mineral intergrowth texture See also * Metaplectic group * Symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calque
In linguistics, a calque () or loan translation is a word or phrase borrowed from another language by literal word-for-word or root-for-root translation. When used as a verb, "to calque" means to borrow a word or phrase from another language while translating its components, so as to create a new word or phrase ( lexeme) in the target language. For instance, the English word ''skyscraper'' has been calqued in dozens of other languages, combining words for "sky" and "scrape" in each language, as for example in German, in Portuguese, in Dutch, in Spanish, in Italian, in Turkish, and ''matenrō'' in Japanese. Calques, like direct borrowings, often function as linguistic gap-fillers, emerging when a language lacks existing vocabulary to express new ideas, technologies, or objects. This phenomenon is widespread and is often attributed to the shared conceptual frameworks across human languages. Speakers of different languages tend to perceive the world through common categori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Category
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds ''L'' into M \times N^, where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus). The notion was introduced by Alan Weinstein, according to whom "Quantization problemsHe means geometric quantization. suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms." The composition of canonical relations is given by a fiber product. Strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions. References ;Notes ;Sources * Further reading * Victor Guillemin and Shlomo Sternberg, ''Some problems in integral geometry and some related problems in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. In the simplest case, these are differential operators. Let F be a field, and let F /math> be the ring of polynomials in one variable with coefficients in F. Then the corresponding Weyl algebra consists of differential operators of form : f_m(x) \partial_x^m + f_(x) \partial_x^ + \cdots + f_1(x) \partial_x + f_0(x) This is the first Weyl algebra A_1. The ''n''-th Weyl algebra A_n are constructed similarly. Alternatively, A_1 can be constructed as the quotient of the free algebra on two generators, ''q'' and ''p'', by the ideal generated by ( ,q- 1). Similarly, A_n is obtained by quotienting the free algebra on ''2n'' generators by the ideal generated by ( _i,q_j- \delta_), \quad \forall i, j = 1, \dots, nwhere \delta_ is the K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Symplectic geometry 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. The term "symplectic", introduced by Hermann Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name " symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Integrator
In mathematics, a symplectic integrator (SI) is a Numerical ordinary differential equations, numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, particle accelerator, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Introduction Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read \dot p = -\frac \quad\mbox\quad \dot q = \frac, where q denotes the position coordinates, p the momentum coordinates, and H is the Hamiltonian. The set of position and momentum coordinates (q,p) are called canonical coordinates. (See Hamiltonian mechanics for more background.) The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form dp \wedge dq. A numerical sche ... [...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 Matrix
In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be extended to 2n\times 2n matrices with entries in other fields, such as the complex numbers, finite fields, ''p''-adic numbers, and function fields. Typically \Omega is chosen to be the block matrix \Omega = \begin 0 & I_n \\ -I_n & 0 \\ \end, where I_n is the n\times n identity matrix. The matrix \Omega has determinant +1 and its inverse is \Omega^ = \Omega^\text = -\Omega. Properties Generators for symplectic matrices Every symplectic matrix has determinant +1, and the 2n\times 2n symplectic matrices with real entries form a subgroup of the general linear group \mathrm(2n;\mathbb) under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form :\omega\colon V\times V \to \mathbb F where F is the field of scalars. A representation of a group ''G'' preserves ''ω'' if :\omega(g\cdot v,g\cdot w)= \omega(v,w) for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a Lie algebra g preserves ''ω'' if :\omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0 for all ''ξ'' in g and ''v'', ''w'' in ''V''. Thus a representation of ''G'' or g is equivalently a group or Lie algebra homomorphism from ''G'' or g to the symplectic group Sp(''V'',''ω'') or its Lie algebra sp(''V'',''ω'') If ''G'' is a compact group (for example, a finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise w ... [...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] |
Symplectic Bone
The skull, or cranium, is typically a bony enclosure around the brain of a vertebrate. In some fish, and amphibians, the skull is of cartilage. The skull is at the head end of the vertebrate. In the human, the skull comprises two prominent parts: the neurocranium and the facial skeleton, which evolved from the first pharyngeal arch. The skull forms the frontmost portion of the axial skeleton and is a product of cephalization and vesicular enlargement of the brain, with several special senses structures such as the eyes, ears, nose, tongue and, in fish, specialized tactile organs such as barbels near the mouth. The skull is composed of three types of bone: cranial bones, facial bones and ossicles, which is made up of a number of fused flat and irregular bones. The cranial bones are joined at firm fibrous junctions called sutures and contains many foramina, fossae, processes, and sinuses. In zoology, the openings in the skull are called fenestrae, the most prominent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
