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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] |
Theory Of Relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old Classical mechanics, theory of mechanics created primarily by Isaac Newton. It introduced concepts including 4-dimensional spacetime as a unified entity of space and time in physics, time, relativity of simultaneity, kinematics, kinematic and gravity, gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytical Mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is ''vectorial mechanics''. By contrast, analytical mechanics uses '' scalar'' properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Analytical mec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...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] |
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] |
Free Motion Equation
A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a reference frame. In non-autonomous mechanics on a configuration space Q\to \mathbb R, a free motion equation is defined as a second order non-autonomous dynamic equation on Q\to \mathbb R which is brought into the form : \overline q^i_=0 with respect to some reference frame (t,\overline q^i) on Q\to \mathbb R. Given an arbitrary reference frame (t,q^i) on Q\to \mathbb R, a free motion equation reads : q^i_=d_t\Gamma^i +\partial_j\Gamma^i(q^j_t-\Gamma^j) - \frac\frac(q^j_t-\Gamma^j) (q^k_t-\Gamma^k), where \Gamma^i=\partial_t q^i(t,\overline q^j) is a connection on Q\to \mathbb R associates with the initial reference frame (t,\overline q^i). The right-hand side of this equation is treated as an inertial force. A free motion equation need not exist in general. It can be defined if and ... [...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] |
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, the ... [...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] |
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, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, '' Mécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair (M,L) consisting of a configuration space M and a smooth function L within that space called a ''Lagrangian''. By convention, L = T - V, where T and V are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Introduction Suppose there exists a bead sliding around on a wire, or a swinging simple p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Manifold
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 ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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