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Dirac Structure
In mathematics a Dirac structure is a geometric construction generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein. In more detail, let ''V'' be a real vector space, and ''V*'' its dual. A (linear) ''Dirac structure'' on ''V'' is a linear subspace ''D'' of V\times V^* satisfying * for all (v,\alpha)\in D one has \left\langle\alpha,\,v\right\rangle=0, * ''D'' is maximal with respect to this property. In particular, if ''V'' is finite dimensional then the second criterion is satisfied if \dim D = \dim V. (Similar definitions can be made for vector spaces over other fields.) An alternative (equivalent) definition often used is that D satisfies D=D^\perp, where orthogonality is with respect to the symmetric bilinear form on V\times V^* given by \bigl\langle(u,\alpha),\,(v,\beta)\bigr\rangle = \left\langle\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Structure
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 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, hence on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Structure
In differential 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 : \ = \h + g \ . Equivalently, \ defines a Lie algebra structure on the vector space (M) of smooth functions on M such that X_:= \: (M) \to (M) is a vector field for each smooth function f (making (M) into a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their ear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University and the University of Miami, and a 1933 Nobel Prize recipient. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics. Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodore James Courant
Theodore James "Ted" Courant is an American mathematician who has conducted research in the fields of differential geometry and classical mechanics. In particular, he made seminal contributions to the study of Dirac manifolds,Courant, Theodore James, ''Dirac manifolds'', Trans. Amer. Math. Soc., 319:631-661, (1990). which generalize both symplectic manifolds and Poisson manifolds, and are related to the Dirac theory of constraints in physics. Some mathematical objects in this field have since been named after him, including the Courant bracket and Courant algebroid. He received his B.A. degree from Reed College and his Ph.D. from The University of California, Berkeley, where he was a student of Alan Weinstein Alan David Weinstein (17 June, 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry. Education and career Weinstein ob .... Ted Courant is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Weinstein
Alan David Weinstein (17 June, 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry. Education and career Weinstein obtained a bachelor's degree at the Massachusetts Institute of Technology in 1964. He received a PhD at University of California, Berkeley in 1967 under the direction of Shiing-Shen Chern. His dissertation was entitled "''The cut locus and conjugate locus of a Riemannian manifold''". He worked then at MIT on 1967 (as Moore instructor) and at Bonn University in 1968/69. In 1969 he became assistant professor at Berkeley, and from 1976 he is full professor. During 1978/79 he was visiting professor at Rice University. Weinstein was awarded in 1971 a Sloan Research Fellowship and in 1985 a Guggenheim Fellowship. In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki. In 1992 he was elected Fellow of the A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (differential Geometry)
In differential geometry, a discipline within mathematics, a distribution on a manifold M is an assignment x \mapsto \Delta_x \subseteq T_x M of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle TM. Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, e.g. integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology, etc. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis. Definition Let M be a smooth manifold; a (smooth) distribution \Delta assigns to any point x \in M a vector subspace \Delta_x \subset T_xM in a smooth way. More precisely, \Delta consists in a collection \_ of vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Form
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. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to 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 form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonholonomic Constraints
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of Newtonian mechanics. Details More precisely, a nonholonomic system, also called an ''anholonomic'' system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering and mechanical engineering, but also in other complex fields such as meteorology. Historically, thermodynamics developed out of a desire to increase the efficiency of early steam engines, particularly through the work of French physicist Sadi Carnot (1824) who believed that engine efficiency was the key that could help France win the Napoleonic Wars. Scots-Irish physicist Lord Kelvin was the first to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Geometry And Physics
The ''Journal of Geometry and Physics'' is a scientific journal in mathematical physics. Its scope is to stimulate the interaction between geometry and physics by publishing primary research and review articles which are of common interest to practitioners in both fields. The journal is published by Elsevier since 1984. The Journal covers the following areas of research: ''Methods of:'' * Algebraic and Differential Topology * Algebraic Geometry * Real and Complex Differential Geometry * Riemannian and Finsler Manifolds * Symplectic Geometry * Global Analysis, Analysis on Manifolds * Geometric Theory of Differential Equations * Geometric Control Theory * Lie Groups and Lie Algebras * Supermanifolds and Supergroups * Discrete Geometry * Spinors and Twistors ''Applications to:'' * Strings and Superstrings * Noncommutative Topology and Geometry * Quantum Groups * Geometric Methods in Statistics and Probability * Geometry Approaches to Thermodynamics * Classical and Quantum Dynam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
