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Encyclopedia > Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...


The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. It has been suggested that geodetic system be merged into this article or section. ... This article is about Earth as a planet. ... An open surface with X-, Y-, and Z-contours shown. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ...

Contents

Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve, and then minimizing this length using standard techniques of calculus and differential equations. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the resulting shape of the band is a geodesic. An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... A rubber band (in some regions known as a binder and in others as an elastic) is a short length of rubber formed in the shape of a loop. ...


Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In physics, geodesics describe the motion of point particles; in particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all described by geodesics in the theory of general relativity. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... A point particle is an idealized particle heavily used in physics. ... An Earth observation satellite, ERS 2 For other uses, see Satellite (disambiguation). ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. ...


This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds. The article geodesic (general relativity) discusses the special case of general relativity in greater detail. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ...


Examples

The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. If A and B are antipodal points (like the North pole and the South pole), then there are infinitely many shortest paths between them. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A sphere is a perfectly symmetrical geometrical object. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ...


Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: IM from the unit interval I to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any tI there is a neighborhood J of t in I such that for any t1, t2J we have In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...

d(gamma(t_1),gamma(t_2))=v|t_1-t_2|.,

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is almost always equipped with natural parametrization, i.e. in the above identity v = 1 and In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

d(gamma(t_1),gamma(t_2))=|t_1-t_2|.,

If the last equality is satisfied for all t1, t2I, the geodesic is called a minimizing geodesic or shortest path.


In general, a metric space may have no geodesics, except constant curves.


(Pseudo-)Riemannian geometry

Just as in a standard metric space, a geodesic on a (pseudo-)Riemannian manifold M is defined as a curve γ(t) minimizes the length of the curve. Explicitly, we can write the length of any curve as In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

l(gamma)=int_gamma sqrt{ pm g(dotgamma(t),dotgamma(t)) },dt ,

where dotgamma represents the derivative with respect to t, and is a vector. The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves. A geodesic, then, is the curve which extremizes this quantity (locally).


In the case of a manifold with torsion-free and metric-compatible connection (which is almost always assumed to be the case in Relativity, for example), a geodesic curve is also an autoparallel curve. That is, the curve parallel transports its own tangent vector, so Torsion of affine connection is a (1,2) tensor given by the formula where is the Lie bracket of the two vector fields. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

at each point along the curve. Here, ∇ stands for the Levi-Civita connection on M. In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...


In this case, using local coordinates on M, we can write the geodesic equation (using the summation convention) as Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

frac{d^2x^lambda }{dt^2} + Gamma^{lambda}_{~mu nu }frac{dx^mu }{dt}frac{dx^nu }{dt} = 0 ,

where xμ(t) are the coordinates of the curve γ(t) and Gamma^{lambda }_{~mu nu } are the Christoffel symbols. This is just an ordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physics a free particle is a particle that is never under the influence of an external force Classical Free Particle The classical free particle is characterized simply by a fixed velocity. ...


Geodesics can also be defined as extremal curves for the following action functional Stationary points (red pluses) and inflection points (green circles). ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

where g is a Riemannian (or pseudo-Riemannian) metric. In pure mathematics, this quantity would generally be referred to as an energy. The geodesic equation can then be obtained as the Euler-Lagrange equations of motion for this action. In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...


In a similar manner, one can obtain geodesics as a solution of the Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian. See Riemannian manifolds in Hamiltonian mechanics for further details. In physics and mathematics, the Hamilton–Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...


Existence and uniqueness

The local existence and uniqueness theorem for geodesics states that geodesics exist, and are unique; this is a variant of the Frobenius theorem. More precisely: In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ...

For any point p in M and for any vector V in TpM (the tangent space to M at p) there exists a unique geodesic gamma , : IM such that
gamma(0) = p , and
dotgamma(0) = V,
where I is a maximal open interval in R containing 0.

In general, I may not be all of R as for example for an open disc in R². The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the Picard-Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and V. The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...


Geodesic flow

Geodesic flow is an mathbb R-action on tangent bundle T(M) of a manifold M defined in the following way flOw is a Flash game created by Jenova Chen and Nicholas Clark. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

G^t(V)=dotgamma_V(t)

where tin mathbb R, Vin T(M) and γV denotes the geodesic with initial data dotgamma_V(0)=V.


It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian. In particular it preserves the (pseudo-)Riemannian metric g, i.e. In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ... In physics, Hamiltonian has distinct but closely related meanings. ...

g(Gt(V),Gt(V)) = g(V,V).

That makes possible to define geodesic flow on unit tangent bundle UT(M) of the Riemannian manifold M. In mathematics, the unit tangent bundle of a Riemannian manifold (M, g), denoted by UT(M) or simply UTM, is a fiber bundle given by the disjoint union where Tx(M) denotes the tangent space to M at x. ...


Geodesic spray

The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray. In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...


See also

This article is on the minimal body of mathematics necessary to understand general relativity. ... In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... A geodesic dome is an almost spherical structure based on a network of struts arranged on great circles (geodesics) lying approximately on the surface of a sphere. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ... In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler-Lagrange equations of motion. ... If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ... If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ... In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic in a Riemannian manifold. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... Sir Barnes Neville Wallis Sir Barnes Neville Wallis, CBE, FRS, RDI, commonly known as Barnes Wallis, (September 26, 1887 – October 30, 1979) was an English scientist, engineer and inventor. ... R100 moored in Saint-Hubert The HM Airship R100 was a rigid airship, the successful private counterpart to the British government R101 project, in a competition intended to maximize innovation. ... This article is in need of attention from an expert on the subject. ... The Vickers Wellesley was a 1930s light bomber built by Vickers-Armstrong Ltd for the Royal Air Force. ... The Vickers Wellington was a twin-engine, medium bomber designed in the mid-1930s at Brooklands in Weybridge, Surrey, by Vickers-Armstrongs Chief Designer, R.K. Pierson. ...

References

  • Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 See section 1.4.
  • Ronald Adler, Maurice Bazin, Menahem Schiffer, Introduction to General Relativity (Second Edition), (1975) McGraw-Hill New York, ISBN 0-07-000423-4 See chapter 2.
  • Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 2.7.
  • Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972) John Wiley & Sons, New York ISBN 0-471-92567-5 See chapter 3.
  • Lev D. Landau and Evgenii M. Lifschitz, The Classical Theory of Fields, (1973) Pergammon Press, Oxford ISBN 0-08-018176-7 See section 87.
  • Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.
  • Tomás Ortín, Gravity and Strings, (2004) Cambridge University Press, New York. Note especially pages 7 and 10.

Ralph H. Abraham (born July 4, 1936) is an American mathematician. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included...

External links


  Results from FactBites:
 
Geodesic - definition of Geodesic in Encyclopedia (678 words)
In mathematics, a geodesic is a generalization of the notion of a "staight line" to curved spaces.
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer.
Geodesics are important in the theory of general relativity where they represent the paths of particles in free fall, that is, particles moving under the influence of no forces except gravity.
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