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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] |
Autonomous System (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...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] |
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] |
Jet Bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions. Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing ''geometrically'' with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.) Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Differential
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Ja ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60) Formal definition Let \overline\pi:\overline Y\to X be a vector bundle with a typical fiber a vector space \overline F. An affine bundle modelled on a vector bundle \overline\pi:\overline Y\to X is a fiber bundle \pi:Y\to X whose typical fiber F is an affine space modelled on \overline F so that the following conditions hold: (i) Every fiber Y_x of Y is an affine space modelled over the corresponding fibers \overline Y_x of a vector bundle \overline Y. (ii) There is an affine bundle atlas of Y\to X whose local trivializations morphisms and transition functions are affine isomorphisms. Dealing with affine bundles, one uses only affine bundle coordinates (x^\mu,y^i) possessing affine transition functions : y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu). There are the bundle morphisms : Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, ... [...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] |
Autonomous System (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...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] |
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] |
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] |
