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In mathematics, a geodesic [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 an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it.

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.

Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles.

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 generally metric geometry. In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. 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. 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.

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.

- $d(gamma(t\_1),gamma(t\_2))=v|t\_1-t\_2|.,$

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 of rectifiable paths, although this minimizing sequence need not converge to a geodesic.

- $nabla\_\{dotgamma\}\; dotgamma=\; 0$

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

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

Geodesics for a (pseudo-)Riemannian manifold M are defined to be geodesics for its Levi-Civita connection. In a Riemannian manifold a geodesic is the same as a curve that locally minimizes the length

- $l(gamma)=int\_gamma\; sqrt\{\; g(dotgamma(t),dotgamma(t))\; \},dt\; ,$

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

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.

- For any point p in M and for any vector V in T
_{p}M (the tangent 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 maximal open 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 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.

- $G^t(V)=dotgamma\_V(t)$

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.

- $g(G^t(V),G^t(V))=g(V,V)$.

- $nabla\_\{dot\{gamma\}(t)\}dot\{gamma\}(t)\; =\; 0.$

This equation is invariant under affine reparameterizations; that is, parameterizations of the form

- $tmapsto\; at+b$

An affine connection is determined by its family of affinely parameterized geodesics, up to torsion . 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 $nabla,\; bar\{nabla\}$ are two connections such that the difference tensor

- $D(X,Y)\; =\; nabla\_XY-bar\{nabla\}\_XY$

Geodesics without a particular parameterization are described by a projective connection.

- 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 the Vickers Wellesley and Vickers Wellington aircraft, and the R100 airship.

- . See chapter 2.
- . See section 2.7.
- . See section 1.4.
- . See section 87.
- . Note especially pages 7 and 10.
- . See chapter 3.

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

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Last updated on Wednesday October 08, 2008 at 20:09:55 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 08, 2008 at 20:09:55 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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