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geodesic
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{{about||geodesics on the Earth|Geodesics on an ellipsoid|geodesics in general relativity|Geodesics in general relativity|other uses|Geodesic (disambiguation)}}File:Spherical triangle.svg|thumb|right|150px|A geodesic triangle on the sphere.The geodesics are great circlegreat circle{{Geodesy}}In differential geometry, a geodesic ({{IPAc-en|ˌ|dʒ|iː|ə|ˈ|d|ɛ|s|ɪ|k|,_|ˌ|dʒ|iː|oʊ|-|,_|-|ˈ|d|iː|-|,_|-|z|ɪ|k}}{{refn|WEB,weblink geodesic – definition of geodesic in English from the Oxford dictionary, OxfordDictionaries.com, 2016-01-20, }}{{refn|{{MerriamWebsterDictionary|geodesic}}}}) is a curve representing in some sense the shortest path between two points in a surface, or more generally in a Riemannian manifold. It is a generalization of the notion of a "straight line" to a more general setting. 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. For a spherical Earth, it is 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.In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

Introduction

The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f(s) to f(t) along the curve equals |st|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization).{{citation needed|date=May 2018}} 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.It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic.A contiguous segment of a geodesic is again a geodesic.In general, 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 parameterized with "constant speed". 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 â†’ t2 from the unit interval on the real number line 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 general relativity, geodesics in spacetime 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 spacetime. 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.

Examples

File:Transpolar geodesic on a triaxial ellipsoid case A.svg|thumb|right|200px|A geodesic on a triaxial ellipsoid.]](File:Insect on a torus tracing out a non-trivial geodesic.gif|thumb|right|If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.)The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points, then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve {{nowrap|γ : I → M}} from an interval I of the reals to the metric space M is a geodesic if there is a constant {{nowrap|v ≥ 0}} such that for any {{nowrap|t ∈ I}} there is a neighborhood J of t in I such that for any {{nowrap|t1, t2 ∈ J}} we have
d(gamma(t_1),gamma(t_2)) = v left| t_1 - t_2 right| .
This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization, i.e. in the above identity v = 1 and
d(gamma(t_1),gamma(t_2)) = left| t_1 - t_2 right| .
If the last equality is satisfied for all {{nowrap|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.

Riemannian geometry

In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ : [a,b] â†’ M is defined by
L(gamma)=int_a^b sqrt{ g_{gamma(t)}(dotgamma(t),dotgamma(t)) },dt.
The distance d(p, q) between two points p and q of M is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ : [a,b] â†’ M such that γ(a) = p and γ(b) = q. In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the following action or energy functional
E(gamma)=frac{1}{2}int_a^b g_{gamma(t)}(dotgamma(t),dotgamma(t)),dt.
Note that all minima of E are also minima of L, but L is a bigger set since paths that are minima of L can be arbitrarily re-parameterized, while minima of E cannot.For a piecewise C^1 curve (more generally, a W^{1,2} curve), the Cauchy–Schwarz inequality gives
L(gamma)^2 le 2(b-a)E(gamma)
with equality if and only if g(gamma',gamma') is equal to a constant a.e. It happens that minimizers of E(gamma) also minimize L(gamma), because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of gamma, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of the functional L(gamma) are generally not very regular, because arbitrary reparameterizations are allowed.The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by
frac{d^2x^lambda }{dt^2} + Gamma^{lambda}_{mu nu }frac{dx^mu }{dt}frac{dx^nu }{dt} = 0,
where Gamma^lambda_{munu} are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.

Calculus of variations

Techniques of the classical calculus of variations can be applied to examine the energy functional E. The first variation of energy is defined in local coordinates by
delta E(gamma)(varphi) = left.frac{partial}{partial t}right|_{t=0} E(gamma + tvarphi).
The critical points of the first variation are precisely the geodesics. The second variation is defined by
delta^2 E(gamma)(varphi,psi) = left.frac{partial^2}{partial s , partial t} right|_{s=t=0} E(gamma + tvarphi + spsi).
In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields. Jacobi fields are thus regarded as variations through geodesics.By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows. They are solutions of the associated Hamilton equations, with (pseudo-)Riemannian metric taken as Hamiltonian.

Affine geodesics

{{See also|Geodesics in general relativity}}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{{NumBlk|:| nabla_{dotgamma} dotgamma= 0|{{EquationRef|1}}}}at each point along the curve, where dotgamma is the derivative with respect to t. More precisely, in order to define the covariant derivative of dotgamma it is necessary first to extend dotgamma to a continuously differentiable vector field in an open set. However, the resulting value of ({{EquationNote|1}}) is independent of the choice of extension.Using local coordinates on M, we can write the geodesic equation (using the summation convention) as
frac{d^2gamma^lambda }{dt^2} + Gamma^{lambda}_{mu nu }frac{dgamma^mu }{dt}frac{dgamma^nu }{dt} = 0 ,
where gamma^mu = x^mu circ gamma (t) are the coordinates of the curve γ(t) and Gamma^{lambda }_{mu nu } are the Christoffel symbols of the connection ∇. This is 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. Indeed, the equation nabla_{dotgamma} dotgamma= 0 means that the acceleration vector of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by the gravity.

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. 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 , : 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.

Geodesic flow

Geodesic flow is a local R-action on the tangent bundle TM of a manifold M defined in the following way
G^t(V)=dotgamma_V(t)
where t âˆˆ R, V âˆˆ TM and gamma_V denotes the geodesic with initial data dotgamma_V(0)=V. Thus, G^t(V) = exp(tV) is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M.On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical one-form. In particular the flow preserves the (pseudo-)Riemannian metric g, i.e.
g(G^t(V),G^t(V))=g(V,V). ,
In particular, when V is a unit vector, gamma_V remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

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.More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles:
TTM = Hoplus V.
The geodesic spray is the unique horizontal vector field W satisfying
pi_* W_v = v,
at each point v âˆˆ TM; here π∗ : TTM â†’ TM denotes the pushforward (differential) along the projection Ï€ : TM â†’ M associated to the tangent bundle.More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM  {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf. Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy
H_{lambda X} = d(S_lambda)_X H_X,
for every X âˆˆ TM  {0} and λ > 0. Here d(Sλ) is the pushforward along the scalar homothety S_lambda: Xmapsto lambda X. A particular case of a non-linear connection arising in this manner is that associated to a Finsler manifold.

Affine and projective geodesics

Equation ({{EquationNote|1}}) is invariant under affine reparameterizations; that is, parameterizations of the form
tmapsto at+b
where a and b are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of ({{EquationNote|1}}) are called geodesics with affine parameter.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 nabla, bar{nabla} are two connections such that the difference tensor
D(X,Y) = nabla_XY-bar{nabla}_XY
is skew-symmetric, then nabla and bar{nabla} have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as nabla, but with vanishing torsion.Geodesics without a particular parameterization are described by a projective connection.

Computational methods

Efficient solvers for the minimal geodesic problem on surfaces posed as Eikonal equations can be found inR. Kimmel, A. Amir, and A. M. Bruckstein. Finding shortest paths on surfaces using level sets propagation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(6):635–640, 1995.R. Kimmel and J. A. Sethian. Computing Geodesic Paths on Manifolds in the Proceedings of National Academy of Sciences, 95(15):8431–8435, July, 1998.

Applications

{{expand section|date=June 2014}}Geodesics serve as the basis to calculate:

See also

{{div col|colwidth=25em}} {{div col end}}

References

{{Reflist}}
  • {{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}}
{{Commons category|Geodesic (mathematics)}}

Further reading

{{more footnotes|date=July 2014}}
  • {{Citation | last1=Adler | first1=Ronald | last2=Bazin | first2=Maurice | last3=Schiffer | first3=Menahem | title=Introduction to General Relativity | publisher=McGraw-Hill | location=New York | edition=2nd | isbn=978-0-07-000423-8 | year=1975}}. See chapter 2.
  • {{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham (mathematician) | last2=Marsden | first2=Jerrold E. | author2-link=Jerrold E. Marsden | title=Foundations of mechanics | publisher=Benjamin-Cummings | location=London | isbn=978-0-8053-0102-1 | year=1978}}. See section 2.7.
  • {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-3-540-42627-1 | year=2002}}. See section 1.4.
  • {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry|volume=Vol. 1| publisher=Wiley-Interscience | year=1996|edition=New|isbn=0-471-15733-3}}.
  • {{Citation | last1=Landau | first1=L. D. | author1-link=Lev Landau | last2=Lifshitz | first2=E. M. | author2-link=Evgeny Lifshitz | title=Classical Theory of Fields | publisher=Pergamon | location=Oxford | isbn=978-0-08-018176-9 | year=1975}}. See section 87.
  • {{Citation | last1=Misner | first1=Charles W. | author1-link=Charles W. Misner | last2=Thorne | first2=Kip | author2-link=Kip Thorne | last3=Wheeler | first3=John Archibald | author3-link=John Archibald Wheeler | title=Gravitation | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973}}
  • {{Citation | last1=Ortín | first1=Tomás | title=Gravity and strings | publisher=Cambridge University Press | isbn=978-0-521-82475-0 | year=2004}}. Note especially pages 7 and 10.
  • {{springer|first=Yu.A.|last=Volkov|title=Geodesic line|id=G/g044120}}.
  • {{Citation | last1=Weinberg | first1=Steven | author1-link=Steven Weinberg | title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity | publisher=John Wiley & Sons | location=New York | isbn=978-0-471-92567-5 | year=1972}}. See chapter 3.

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