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Sardanashvily
Gennadi Sardanashvily (russian: Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 – September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University. Biography Gennadi Sardanashvily graduated from Moscow State University (MSU) in 1973, he was a Ph.D. student of the Department of Theoretical Physics ( MSU) in 1973–76, where he held a position in 1976. He attained his Ph.D. degree in physics and mathematics from MSU, in 1980, with Dmitri Ivanenko as his supervisor, and his D.Sc. degree in physics and mathematics from MSU, in 1998. Gennadi Sardanashvily was the founder and Managing Editor (2003 - 2013) of the International Journal of Geometric Methods in Modern Physics (IJGMMP). He was a member of Lepage Research Institute (Czech Republic). Research area Gennadi Sardanashvily research area is geometric method in classical and quantum mechanics and field theory, gravitation theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian System
In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of . In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle over the time axis . In particular, if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones. Lagrangians and Euler–Lagrange operators A Lagrangian density (or, simply, a Lagrangian) of order is defined as an -form, , on the -order jet manifold of . A Lagrangian can be introduced as an element of the variational bicomplex of the differential graded algebra of exterior forms on jet manifolds of . The coboundary operator of this bicomplex contains the variational operator which, acting on , defines the associated Euler–Lagrange operator . In coordinates Given bundle coordinates on a fiber bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Field (classical)
Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles. At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group G of a principal bundle P\to X to its closed subgroup H. By the well-known theorem, such a reduction takes place if and only if there exists a global section h of the quotient bundle P/G\to X. This section is treated as a classical Higgs field. A key point is that there exists a composite bundle P\to P/G\to X where P\to P/G is a principal bundle with the structure group H. Then matter fields, possess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Gravitation Theory
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. ''Gauge gravitation theory'' should not be confused with the similarly-named gauge theory gravity, which is a formulation of (classical) gravitation in the language of geometric algebra. Nor should it be confused with Kaluza–Klein theory, where the gauge fields are used to describe particle fields, but not gravity itself. Overview The first gauge model of gravity was suggested by Ryoyu Utiyama (1916–1990) in 1956 just two years after birth of the gauge theory itself. However, the initial attempts to construct the gauge theory of gravity by analogy with the gauge models of internal symmetries encountered a problem of treating general covariant transformations and establishing the gauge status of a pseudo-Riemannian metric (a tetrad field). In order to overcome this drawback, representing tetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitri Ivanenko
Dmitri Dmitrievich Ivanenko (russian: Дми́трий Дми́триевич Иване́нко; July 29, 1904 – December 30, 1994) was a Ukrainian theoretical physicist who made great contributions to the physical science of the twentieth century, especially to nuclear physics, field theory, and gravitation theory. He worked in the Poltava Gravimetric Observatory of the Institute of Geophysics of NAS of Ukraine, was the head of the Theoretical Department Ukrainian Physico-Technical Institute in Kharkiv, Head of the Department of Theoretical Physics of the Kharkiv Institute of Mechanical Engineering. Professor of University of Kharkiv, Professor of Moscow State University (since 1943). Biography Dmitri Ivanenko was born on July 29, 1904 in Poltava, where he finished school, in 1920-1923 he studied at the Poltava Pedagogical Institute and began his creative path as a teacher of physics in middle school. Then D. D. Ivanenko studied at Kharkiv University, from which in 1923 he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-autonomous Mechanics
Non-autonomous mechanics describe non- relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle Q\to \mathbb R over the time axis \mathbb R coordinated by (t,q^i). This bundle is trivial, but its different trivializations Q=\mathbb R\times M correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection \Gamma on Q\to\mathbb R which takes a form \Gamma^i =0 with respect to this trivialization. The corresponding covariant differential (q^i_t-\Gamma^i)\partial_i determines the relative velocity with respect to a reference frame \Gamma. As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory). The variational bicomplex is a cochain complex of the differential graded algebra of exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and Euler–Lagrange operators on a fiber bundle are defined as elements of this bicomplex. Cohomology of the variational bicomplex leads to the global first variational formula and first Noether's theorem. Extended to Lagrangian theory of even and odd fields on graded manifolds, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory. See also * Calculus of variations *Lagrangian system *Jet bundle In differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether Identities
In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian ''L'', Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of ''L''. Any Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian ''L'' is called degenerate if the Euler–Lagrange operator of ''L'' satisfies non-trivial Noether identities. In this case Euler–Lagrange equations are not independent. Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities. Yang–Mills gauge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Graded manifolds A graded manifold of dimension (n,m) is defined as a locally ringed space (Z,A) where Z is an n-dimensional smooth manifold and A is a C^\infty_Z-sheaf of Grassmann algebras of rank m where C^\infty_Z is the sheaf of smooth real functions on Z. The sheaf A is called the structure sheaf of the graded manifold (Z,A), and the manifold Z is said to be the body of (Z,A). Sections of the sheaf A are called graded functions on a graded manifold (Z,A). They make up a graded commutative C^\infty(Z)-ring A(Z) called the structure ring of (Z,A). The well-known B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superintegrable Hamiltonian System
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2n-dimensional symplectic manifold for which the following conditions hold: (i) There exist k>n independent integrals F_i of motion. Their level surfaces (invariant submanifolds) form a fibered manifold F:Z\to N=F(Z) over a connected open subset N\subset\mathbb R^k. (ii) There exist smooth real functions s_ on N such that the Poisson bracket of integrals of motion reads \= s_\circ F. (iii) The matrix function s_ is of constant corank m=2n-k on N. If k=n, this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows. Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold F is a fiber bundle in tori T^m. There exists a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Hamiltonian Field Theory
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. Definition The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field. Since each point mass has one or more degrees of freedom, the field formulation has infinitely many degrees of freedom. One scalar field The Hamiltonian density is the continuous analogue for fields; it is a function of the fields, the conjugate "momentum" fields, and possibly the space and time coordinates themselves. For one scalar field , the Hamiltonian density i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether's Second Theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. :Translated in The action ''S'' of a physical system is an integral of a so-called Lagrangian function ''L'', from which the system's behavior can be determined by the principle of least action. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by ''k'' arbitrary functions and their derivatives up to order ''m'', then the functional derivatives of ''L'' satisfy a system of ''k'' differential equations. Noether's second theorem is sometimes used in gauge theory. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model. The theorem is named after Emmy Noether. See also * Noether's first theorem * Noether identities * Gauge symmetry (mathematics) In mathematics, any Lagrangian sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
