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In mathematics, a geodesic is a generalization of the notion of a "Line (mathematics)" to "manifolds".In presence of a Metric (mathematics), 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 parallel transport 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 line 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.

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.

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 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.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian manifold and pseudo-Riemannian manifolds. The article geodesic (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 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.

Metric geometry In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from the unit interval I to the metric space M is a geodesic if there is a mathematical 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. However, in metric geometry the geodesic considered is almost always equipped with Curve#Length of curves, 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 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.

(Pseudo-)Riemannian geometry Just as in a standard metric space, a geodesic on a (pseudo-Riemannian manifold-)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 l(\gamma)=\int_\gamma \sqrt{ \pm g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ , where \dot\gamma 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 of connection-free and metric tensor connection (mathematics) (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 \dot\gamma^\mu \nabla_\mu \dot\gamma^\nu = 0 at each point along the curve. Here, ∇ stands for the Levi-Civita connection on M.

In this case, 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^\mu (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.

Geodesics can also be defined as Stationary point curves for the following action (physics) 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 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 a similar manner, one can obtain geodesics as a solution of the Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian mechanics. See Hamiltonian mechanics#Riemannian manifolds for further details.

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 (differential topology). More precisely:

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 \, : I → M such that :\gamma(0) = p \, and :\dot\gamma(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 smooth function on both p and V.

Geodesic flow Geodesic flow is an \mathbb R-group 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 \gamma_V denotes the geodesic with initial data \dot\gamma_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. g(G^t(V),G^t(V))=g(V,V). That makes possible to define geodesic flow on unit tangent bundle UT(M) of the Riemannian manifold M.

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.

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 parallel transport along it.

See also

References

External links

In mathematics, a geodesic is a generalization of the notion of a "Line (mathematics)" to "manifolds".In presence of a Metric (mathematics), 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 parallel transport 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 line 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.

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.

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 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.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian manifold and pseudo-Riemannian manifolds. The article geodesic (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 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.

Metric geometry In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from the unit interval I to the metric space M is a geodesic if there is a mathematical 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. However, in metric geometry the geodesic considered is almost always equipped with Curve#Length of curves, 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 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.

(Pseudo-)Riemannian geometry Just as in a standard metric space, a geodesic on a (pseudo-Riemannian manifold-)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 l(\gamma)=\int_\gamma \sqrt{ \pm g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ , where \dot\gamma 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 of connection-free and metric tensor connection (mathematics) (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 \dot\gamma^\mu \nabla_\mu \dot\gamma^\nu = 0 at each point along the curve. Here, ∇ stands for the Levi-Civita connection on M.

In this case, 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^\mu (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.

Geodesics can also be defined as Stationary point curves for the following action (physics) 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 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 a similar manner, one can obtain geodesics as a solution of the Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian mechanics. See Hamiltonian mechanics#Riemannian manifolds for further details.

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 (differential topology). More precisely:

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 \, : I → M such that :\gamma(0) = p \, and :\dot\gamma(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 smooth function on both p and V.

Geodesic flow Geodesic flow is an \mathbb R-group 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 \gamma_V denotes the geodesic with initial data \dot\gamma_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. g(G^t(V),G^t(V))=g(V,V). That makes possible to define geodesic flow on unit tangent bundle UT(M) of the Riemannian manifold M.

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.

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 parallel transport along it.

See also

References

External links



Geodesic - Wikipedia, the free encyclopedia
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".

Geodesic dome - Wikipedia, the free encyclopedia
A geodesic dome is an almost spherical structure based on a network of great circles (geodesics) lying approximately on the surface of a sphere.

Geodesic -- from Wolfram MathWorld
A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines.

Solardome is the UK's leading designer, manufacturer and supplier of ...
Manufacturers of a standard range of fully glazed geodesic domes used as greenhouses, sunrooms, garden rooms and spa/swimming pool covers.

Solardome Geodesic Domes :: How geodesic domes work :: Angle of ...
How do geodesic domes work? Information and animated 3D presentation demonstrating how geodesic domes work provided by Solardome Industries.

Home Page Geodesic
Geodesic develops innovative Mobile Solutions for Information, Communication and Entertainment ... Geodesic shows robust growth in the First Quarter, Total Income up 116% and net ...

Geodesic Domes, geodesic domes, geodomes, geodesic, yoga studio ...
Custom designed and built cotton canvas tents. Albion Canvas produces a huge range of tents with everything from Period tents for reenactors Film and TV through to tipis and Yurts.

Richard Buckminster Fuller
A Geodesic Dome is a particular type of structure shaped like a part of a sphere or a ball. An American engineer called Richard Buckminster Fuller in the 1940's ...

MIMS EPrints - 2006.4: Computing the Geodesic Interpolating Spline
We examine non-rigid image registration by knotpoint matching. We consider registering two images, each with a set of knotpoints marked, where one of the images is to be ...

Flickr: geodesic's Photostream
Guest Passes let you share your photos that aren't public. Anyone can see your public photos anytime, whether they're a Flickr member or not. But!

 

Geodesic



 
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