- Geodesic
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great circle arcs.] Inmathematics , a**geodesic**IPA|/ˌdʒiəˈdɛsɪk, -ˈdisɪk/ [jee-uh-des-ik, -dee-sik] 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 anaffine connection , geodesics are defined to be curves whosetangent vector s remain parallel if they are transported along it.The term "geodesic" comes from "

geodesy ", the science of measuring the size and shape ofEarth ; in the original sense, a geodesic was the shortest route between two points on the Earth'ssurface , namely, a segment of agreat circle . The term has been generalized to include measurements in much more general mathematical spaces; for example, ingraph theory , one might consider a geodesic between two vertices/nodes of a graph.Geodesics are of particular importance in

general relativity , as they describe the motion of inertial test particles.**Introduction**The shortest path between two points in a curved space can be found by writing the

equation for the length of acurve (a function "f" from an open interval of**R**to the manifold), and then minimizing this length using thecalculus of variations . 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 theenergy of the curve; minimizing the energy leads to the same equations for a "constant velocity" geodesic. Intuitively, one can understand this second formulation by noting that anelastic 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.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"→"t"

^{2}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 generallymetric geometry . In relativisticphysics , geodesics describe the motion ofpoint particle s under the influence of gravity alone. In particular, the path taken by a falling rock, an orbitingsatellite , or the shape of aplanetary orbit are all geodesics in curved space-time. More generally, the topic ofsub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and

pseudo-Riemannian manifold s. The articlegeodesic (general relativity) discusses the special case of general relativity in greater detail.**Examples**The most familiar examples are the straight lines in

Euclidean geometry . On asphere , the images of geodesics are thegreat circle s.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" areantipodal point s (like the North pole and the South pole), then there are "infinitely many" shortest paths between them.**Metric geometry**In

metric geometry , a geodesic is a curve which is everywherelocally adistance minimizer. More precisely, acurve γ: "I" → "M" from an interval "I" of the reals to themetric 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 "t"_{1}, "t"_{2}∈ "J" we have:$d(gamma(t\_1),gamma(t\_2))=v|t\_1-t\_2|.,$

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parametrization, i.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 "t"

_{1}, "t"_{2}∈"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 ofrectifiable path s, although this minimizing sequence need not converge to a geodesic.**(Pseudo-)Riemannian geometry**A

**geodesic**on a smooth manifold "M" with an affine connection ∇ is defined as acurve γ("t") such that parallel transport along the curve preserves the tangent vector to the curve, so:$abla\_\{dotgamma\}\; dotgamma=\; 0$at each point along the curve, where $dotgamma$ is the derivative with respect to $t$.Using

local coordinates on "M", we can write the**geodesic equation**(using thesummation convention ) as:$frac\{d^2x^lambda\; \}\{dt^2\}\; +\; Gamma^\{lambda\}\_\{~mu\; u\; \}frac\{dx^mu\; \}\{dt\}frac\{dx^\; u\; \}\{dt\}\; =\; 0\; ,$where $x^mu\; (t)$ are the coordinates of the curve γ("t") and $Gamma^\{lambda\; \}\_\{~mu\; u\; \}$ are theChristoffel symbol s of the connection ∇. This is just anordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view ofclassical mechanics , geodesics can be thought of as trajectories offree particle s in a manifold.Geodesics for a (pseudo-)

Riemannian manifold "M" are defined to be geodesics for itsLevi-Civita connection .In a Riemannian manifold a geodesic is the same as a curve that locally minimizes the length :$l(gamma)=int\_gamma\; sqrt\{\; g(dotgamma(t),dotgamma(t))\; \},dt\; ,$and is parametrized so that the tangent vector has constant length. Geodesics can also be defined as extremal curves for the following action energy functional:$S(gamma)=frac\{1\}\{2\}int\; g(dotgamma(t),dotgamma(t)),dt,$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 theEuler–Lagrange equations of motion for this action.In a similar manner, one can obtain geodesics as a solution of the

Hamilton–Jacobi equation s, with (pseudo-)Riemannian metric taken as Hamiltonian. See Riemannian manifolds in Hamiltonian mechanics for further details.**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. More precisely:

:For any point "p" in "M" and for any vector "V" in "T"

_{"p"}"M" (thetangent space to "M" at "p") there exists a unique geodesic $gamma\; ,$ : "I" → "M" such that ::$gamma(0)\; =\; p\; ,$ and ::$dotgamma(0)\; =\; V$,:where "I" is a maximalopen interval in**R**containing 0.In general, "I" may not be all of

**R**as for example for an open disc in**R**^{2}. The proof of this theorem follows from the theory ofordinary differential equation s, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from thePicard-Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both "p" and "V".**Geodesic flow**Geodesic flow is an $mathbb\; R$-action on

tangent bundle $T(M)$ of a manifold $M$ defined in the following way :$G^t(V)=dotgamma\_V(t)$where $tin\; mathbb\; R$, $Vin\; T(M)$ and $gamma\_V$ 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 theHamiltonian . 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 onunit tangent bundle $UT(M)$ of the Riemannian manifold $M$ when the geodesic $gamma\_V$ is of unit speed.**Geodesic spray**The geodesic flow defines a family of curves in the

tangent bundle . The derivatives of these curves define avector field on thetotal space of the tangent bundle, known as the**geodesic spray**.**Affine and projective geodesics**In the presence of a metric, geodesics are (locally) the length-minimizing curves. However, even if a manifold lacks a metric, geodesics are still well-defined in the presence of an affine connection. A curve in such a manifold is a geodesic if its tangent vector remains parallel to the curve when it is transported along it. Geodesics defined in this way carry a preferred class of

**affine parameterizations**. These are those parameterizations for which:$abla\_\{dot\{gamma\}(t)\}dot\{gamma\}(t)\; =\; 0.$This equation is invariant under affine reparameterizations; that is, parameterizations of the form:$tmapsto\; at+b$where "a" and "b" are constant real numbers.

An affine connection is "determined by" its family of affinely parameterized geodesics, up to torsion harv|Spivak|1999|loc=Chapter 6, Addendum I. The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if $abla,\; ar\{\; abla\}$ are two connections such that the difference tensor:$D(X,Y)\; =\; abla\_XY-ar\{\; abla\}\_XY$is

skew-symmetric , then $abla$ and $ar\{\; abla\}$ have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as $abla$, but with vanishing torsion.Geodesics without a particular parameterization are described by a

projective connection .**ee also**

*Basic introduction to the mathematics of curved spacetime

*Complex geodesic

*Differential geometry of curves

*Exponential map

*Geodesic dome

*Geodesic (general relativity)

*Geodesics as Hamiltonian flows

*Hopf-Rinow theorem

*Intrinsic metric

*Jacobi field

*Quasigeodesic

*Solving the geodesic equations

*Barnes Wallis , who applied geodesics to aircraft structural design in the design of theVickers Wellesley andVickers Wellington aircraft, and theR100 airship.**References***. "See chapter 2".

*. "See section 2.7".

*. "See section 1.4".

*. "See section 87".

*

*. Note especially pages 7 and 10.

*Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=A Comprehensive introduction to differential geometry (Volume 2) | publisher=Publish or Perish | location=Houston, TX | isbn=978-0-914098-71-3 | year=1999

*. "See chapter 3".**External links*** [

*http://www.black-holes.org/relativity5.html Caltech Tutorial on Relativity*] — A nice, simple explanation of geodesics with accompanying animation.

*Wikimedia Foundation.
2010.*