Geodesic

In mathematics, a geodesic /ˌdʒiəˈdɛsɪk, -ˈdisɪk/[jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line" to "curved spaces". Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it. In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold.

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. Geodesy (dʒiːˈɒdɪsi also called geodetics, a branch of Earth sciences, is the scientific discipline that deals EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. 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. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

1 Introduction
- 1.1 Examples
2 Metric geometry
3 (Pseudo-)Riemannian geometry
4 See also
5 References
6 External links

Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. An equation is a mathematical statement, in symbols, that two things are exactly 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 of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. This has some minor technical problems, because there is an infinite dimensional space of different ways to parametrize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parametrized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to|s−t|. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a "constant velocity" geodesic. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός 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. A rubber band (in some regions known as a binder, elastic band, lackey band, elastic blubber, "laggy band" or gumband')

In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parametrized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t→t² from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A point particle (or point-like, often spelled pointlike) is an idealized object heavily used in Physics. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time. This article is about artificial satellites For natural satellites also known as moons see Natural satellite. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star 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 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. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. The article geodesic (general relativity) discusses the special case of general relativity in greater detail. In General relativity, Geodesics generalize the notion of "straight lines" to curved Spacetime.

Examples

The most familiar examples are the straight lines in Euclidean geometry. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. On a sphere, the images of geodesics are the great circles. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. 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. 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

Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods Distance is a numerical description of how far apart objects are More precisely, a curve γ: I → M from an interval I of the reals to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t₁, t₂ ∈ J we have

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

This generalizes the notion of geodesic for Riemannian manifolds. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. However, in metric geometry the geodesic considered is often equipped with natural parametrization, i. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object e. in the above identity v = 1 and

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

If the last equality is satisfied for all t₁, t₂ ∈I, the geodesic is called a minimizing geodesic or shortest path.

In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic. In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

(Pseudo-)Riemannian geometry

A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so

$\nabla_{\dot\gamma} \dot\gamma= 0$

at each point along the curve, where $\dot\gamma$ is the derivative with respect to $t$ . In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

Using local coordinates on M, we can write the geodesic equation (using the summation convention) as

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

where $x μ (t)$ are the coordinates of the curve γ(t) and $\Gamma^{\lambda }_{~\mu \nu }$ are the Christoffel symbols of the connection ∇. Local coordinates are measurement indices into a local Coordinate system or a local Coordinate space. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita This is just an ordinary differential equation for the coordinates. 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 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Physics, a free particle is a particle that in some sense is not bound

Geodesics for a (pseudo-)Riemannian manifold M are defined to be geodesics for its Levi-Civita connection. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i In a Riemannian manifold a geodesic is the same as a curve that locally minimizes the length

$l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,$

and is parametrized so that the tangent vector has constant length. Geodesics can also be defined as extremal curves for the following action functional

$S(\gamma)=\frac{1}{2}\int g(\dot\gamma(t),\dot\gamma(t))\,dt,$

where $g$ is a Riemannian (or pseudo-Riemannian) metric. In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation 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 Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions

In a similar manner, one can obtain geodesics as a solution of the Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian. In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. See Riemannian manifolds in Hamiltonian mechanics for further details. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

Existence and uniqueness

The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique; this is a variant of the Frobenius theorem. In Mathematics, Frobenius' theorem gives Necessary and sufficient conditions for finding a maximal set of independent solutions of an Overdetermined system More precisely:

For any point p in M and for any vector V in T_pM (the tangent space to M at p) there exists a unique geodesic $\gamma \,$ : I → M such that

$\gamma(0) = p \,$ and

$\dot\gamma(0) = V$ ,

where I is a maximal open interval in R containing 0. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

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. 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 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. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

Geodesic flow

Geodesic flow is an $\mathbb R$ -action on tangent bundle $T (M)$ of a manifold $M$ defined in the following way

$G^t(V)=\dot\gamma_V(t)$

where $t\in \mathbb R$ , $V\in T(M)$ and $γ V$ denotes the geodesic with initial data $\dot\gamma_V(0)=V$ . In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the

It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian. In Mathematics and Physics, a Hamiltonian vector field on a Symplectic manifold is a Vector field, defined for any energy function In particular it preserves the (pseudo-)Riemannian metric $g$ , i. e.

g (G t (V), G t (V)) = g (V, V)

That makes possible to define geodesic flow on unit tangent bundle $U T (M)$ of the Riemannian manifold $M$ when the geodesic $γ V$ is of unit speed. In Mathematics, the unit tangent bundle of a Finsler manifold ( M, ||.

Geodesic spray

The geodesic flow defines a family of curves in the tangent bundle. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space.

References

Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1 . Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic See section 1. 4.
Adler, Ronald; Bazin, Maurice & Schiffer, Menahem (1975), Introduction to General Relativity (2nd ed. ), New York: McGraw-Hill, ISBN 978-0-07-000423-8 . The McGraw-Hill Companies Inc, ( is a Publicly traded corporation headquartered in Rockefeller Center in New York City. See chapter 2.
Abraham, Ralph H. & Marsden, Jerrold E. (1978), Foundations of mechanics, London: Benjamin-Cummings, ISBN 978-0-8053-0102-1 . Ralph H Abraham (b July 4 1936, Burlington Vermont) is an American Mathematician. Jerrold Eldon Marsden ( August 17, 1942 in Ocean Falls, British Columbia, Canada) is a well-known Mathematician. See section 2. 7.
Weinberg, Steven (1972), Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, New York: John Wiley & Sons, ISBN 978-0-471-92567-5 . Steven Weinberg (born May 3, 1933) is an American Physicist, and Nobel laureate in Physics for his contributions with Abdus Salam John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors See chapter 3.
Landau, L. D. & Lifshitz, E. M. (1975), Classical Theory of Fields, Oxford: Pergamon, ISBN 978-0-08-018176-9 . Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 Evgeny Mikhailovich Lifshitz (Евгений Михайлович Лифшиц February 21 1915 &ndash October 29 1985) was a leading Soviet See section 87.
Misner, Charles W.; Thorne, Kip & Wheeler, John Archibald (1973), Gravitation, W. Charles W Misner is an American physicist and one of the authors of " Gravitation " Kip Stephen Thorne (born June 1, 1940) is an American theoretical physicist, known for his prolific contributions in gravitation physics John Archibald Wheeler ( July 9, 1911 &ndash April 13, 2008) was an eminent American Theoretical physicist. In Physics, Gravitation is a very important reference book on Einstein 's theory of gravity by Charles W H. Freeman, ISBN 978-0-7167-0344-0
Ortín, Tomás (2004), Gravity and strings, Cambridge University Press, ISBN 978-0-521-82475-0 . Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Note especially pages 7 and 10.

External links

Caltech Tutorial on Relativity — A nice, simple explanation of geodesics with accompanying animation.
Geomath

Dictionary

-noun

(mathematics) the shortest line between two points on a specific surface

-adjective

of or relating to geodesy
of or relating to a geodesic dome

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