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Spray (mathematics)
In differential geometry, a spray is a vector field ''H'' on the tangent bundle ''TM'' that encodes a quasilinear second order system of ordinary differential equations on the base manifold ''M''. Usually a spray is required to be homogeneous in the sense that its integral curves ''t''→ΦHt(ξ)∈''TM'' obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive reparameterizations. If this requirement is dropped, ''H'' is called a semispray. Sprays arise naturally in Riemannian and Finsler geometry as the geodesic sprays whose integral curves are precisely the tangent curves of locally length minimizing curves. Semisprays arise naturally as the extremal curves of action integrals in Lagrangian mechanics. Generalizing all these examples, any (possibly nonlinear) connection on ''M'' induces a semispray ''H'', and conversely, any semispray ''H'' induces a torsion-free nonlinear connection on ''M''. If the original connection is torsion-free it coincides with the connection induced by ''H'', a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Tangent Bundle
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space ''TM'' of the tangent bundle of a smooth manifold ''M'' . A note on notation: in this article, we denote projection maps by their domains, e.g., ''π''''TTM'' : ''TTM'' → ''TM''. Some authors index these maps by their ranges instead, so for them, that map would be written ''π''''TM''. The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle. Secondary vector bundle structure and canonical flip Since is a vector bundle in its own right, its tangent bundle has the secondary vector bundle structure where is the push-forward of the canonical projection In the following we denote : \xi = \xi^k\frac\Big, _x\in T_xM, \qquad X = X^k\frac\Big, _x\in T_xM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Damodar Dharmananda Kosambi
Damodar Dharmananda Kosambi (31 July 1907 – 29 June 1966) was an Indian polymath with interests in mathematics, statistics, philology, history, and genetics. He contributed to genetics by introducing the ''Kosambi map function''. In statistics, he was the first person to develop orthogonal infinite series expressions for stochastic processes via the Kosambi–Karhunen–Loève theorem. He is also well known for his work in numismatics and for compiling critical editions of ancient Sanskrit texts. His father, Dharmananda Damodar Kosambi, had studied ancient Indian texts with a particular emphasis on Buddhism and its literature in the Pali language. Damodar Kosambi emulated him by developing a keen interest in his country's ancient history. He was also a Marxist historian specialising in ancient India who employed the historical materialist approach in his work. He is particularly known for his classic work '' An Introduction to the Study of Indian History''. He is described ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kosambi Biderivative Operator
Kosambi (Pali) or Kaushambi (Sanskrit) was an important city in ancient India. It was the capital of the Vatsa kingdom, one of the sixteen mahajanapadas. It was located on the Yamuna River about southwest of its confluence with the Ganges at Prayaga (modern Prayagraj). History During 2nd millennium BCE Ochre Coloured Pottery culture spread in the region. Kosambi was one of the greatest cities in India from the late Vedic period until the end of Maurya Empire with occupation continuing until the Gupta Empire. As a small town, it was established in the late Vedic period, by the rulers of Kuru Kingdom as their new capital. The initial Kuru capital Hastinapur was destroyed by floods, and the Kuru King transferred his entire capital with the subjects to a new capital that he built near the Ganga-Jamuma confluence, which was 56 km away from the southernmost part of the Kuru Kingdom now as Prayagraj previously called Allahabad.During the period prior the Maurya Empire, Kosambi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Covariant Derivative
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun ''geodesic'' and the adjective ''geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finsler Manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve as :L(\gamma) = \int_a^b F\left(\gamma(t), \dot(t)\right)\,\mathrmt. Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products. Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them. named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation . Definition A Finsler manifold is a differentiable manifold together with a Finsler metric, which is a continuous nonnegative function defined on the tangent bundle so that for each point of , * for every two vectors tangent to at (subadditivity). * for all (but not necessarily for& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Vector Field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions ''f'' and ''g'' on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of ''f'' and ''g''. Definition Suppose that is a symplectic ma ... [...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 even sinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Tangent Bundle
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space ''TM'' of the tangent bundle of a smooth manifold ''M'' . A note on notation: in this article, we denote projection maps by their domains, e.g., ''π''''TTM'' : ''TTM'' → ''TM''. Some authors index these maps by their ranges instead, so for them, that map would be written ''π''''TM''. The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle. Secondary vector bundle structure and canonical flip Since is a vector bundle in its own right, its tangent bundle has the secondary vector bundle structure where is the push-forward of the canonical projection In the following we denote : \xi = \xi^k\frac\Big, _x\in T_xM, \qquad X = X^k\frac\Big, _x\in T_xM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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