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Relativistic System (mathematics)
In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle Q\to \mathbb R over \mathbb R. For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold Q whose fibration over \mathbb R is not fixed. Such a system admits transformations of a coordinate t on \mathbb R depending on other coordinates on Q. Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space Q= \mathbb R^4 is of this type. Since a configuration space Q of a relativistic system has no preferable fibration over \mathbb R, a velocity space of relativistic system is a first order jet manifold J^1_1Q of one-dimensional submanifolds of Q. The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-autonomous System (mathematics)
In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle Q\to \mathbb R over \mathbb R. For instance, this is the case of non-autonomous mechanics. An ''r''-order differential equation on a fiber bundle Q\to \mathbb R is represented by a closed subbundle of a jet bundle J^rQ of Q\to \mathbb R. A dynamic equation on Q\to \mathbb R is a differential equation which is algebraically solved for a higher-order derivatives. In particular, a first-order dynamic equation on a fiber bundle Q\to \mathbb R is a kernel of the covariant differential of some connection \Gamma on Q\to \mathbb R. Given bundle coordinates (t,q^i) on Q and the adapted coordinates (t,q^i,q^i_t) on a first-order jet manifold J^1Q, a first-order dynamic equation reads : q^i_t=\Gamma (t,q^i). For instance, this is the case of Hamiltonian non-autonomous mechanics. A second-order dynamic equation : q^i_=\xi^i(t,q^j,q^j_t) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Classical Field Theory
In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis ℝ. Examples Many important examples of classical field theories which are of interest in quantum field theory are given below. In particular, these are the theories which make up the Standard model of particle physics. These examples will be used in the discussion of the general mathematical formulation of classical field theory. Uncoupled theories * Scala ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi 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. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is, frames of reference with no acceleration). # The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer. Origins and significance Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".Albert Einstein (1905)''Zur Elektrodynamik bewegter Körper'', ''Annalen der Physik'' 17: 891; English translatioOn the Electrodynamics of Moving Bodiesby George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920). The incompa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relativistic Mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light ''c''. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at ''any'' speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics. As with classical mechanics, the subject can be divided into "kin ... [...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] |
Non-autonomous System (mathematics)
In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle Q\to \mathbb R over \mathbb R. For instance, this is the case of non-autonomous mechanics. An ''r''-order differential equation on a fiber bundle Q\to \mathbb R is represented by a closed subbundle of a jet bundle J^rQ of Q\to \mathbb R. A dynamic equation on Q\to \mathbb R is a differential equation which is algebraically solved for a higher-order derivatives. In particular, a first-order dynamic equation on a fiber bundle Q\to \mathbb R is a kernel of the covariant differential of some connection \Gamma on Q\to \mathbb R. Given bundle coordinates (t,q^i) on Q and the adapted coordinates (t,q^i,q^i_t) on a first-order jet manifold J^1Q, a first-order dynamic equation reads : q^i_t=\Gamma (t,q^i). For instance, this is the case of Hamiltonian non-autonomous mechanics. A second-order dynamic equation : q^i_=\xi^i(t,q^j,q^j_t) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jet (mathematics)
In mathematics, the jet is an operation that takes a differentiable function ''f'' and produces a polynomial, the truncated Taylor polynomial of ''f'', at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces. It concludes with a description of jets between manifolds, and how these jets can be constructed intrinsically. In this more general context, it summarizes some of the applications of jets to differential geometry and the theory of differential equations. Jets of functions between Euclidean spaces Before giving a rigorous definition of a jet, it is useful to examine some special cases. One-dimensional cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.This makes spacetime distance an invariant. Becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is, frames of reference with no acceleration). # The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer. Origins and significance Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".Albert Einstein (1905)''Zur Elektrodynamik bewegter Körper'', ''Annalen der Physik'' 17: 891; English translatioOn the Electrodynamics of Moving Bodiesby George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920). The incompa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
