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| {{for|geodesics on the [[Earth]] and other [[ellipsoid]]s|Geodesics on an ellipsoid}}
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| {{Geodesy}}
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| [[File:Spherical triangle.svg|thumb|right|150px|A geodesic triangle on the sphere.
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| The geodesics are [[great circle]] arcs.]]
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| In [[mathematics]], particularly [[differential geometry]], a '''geodesic''' ({{IPAc-en|ˌ|dʒ|iː|ɵ|ˈ|d|iː|z|ɨ|k}} {{respell|JEE|o|DEE|zik}} or {{IPAc-en|ˌ|dʒ|iː|ɵ|ˈ|d|ɛ|s|ɨ|k}} {{respell|JEE|o|DES|ik}}) is a generalization of the notion of a "[[Line (mathematics)|straight line]]" to "[[manifold|curved space]]s". In the presence of an [[affine connection]], a geodesic is defined to be a curve whose [[Tangent space|tangent vector]]s remain parallel if they are [[parallel transport|transported]] along it. If this connection is the [[Levi-Civita connection]] induced by a [[Riemannian metric]], then the geodesics are ([[local property|locally]]) the shortest path between points in the space.
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| 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|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 [[vertex (graph theory)|vertices]]/nodes of a [[graph (mathematics)|graph]].
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| Geodesics are of particular importance in [[general relativity]]. [[Geodesics in general relativity]] describe the motion of inertial [[test particles]].
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| ==Introduction==
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| 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]]. 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 demand not only that the curve locally minimize length but also that it is parameterized "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 geodesic (here "constant velocity" is a consequence of minimisation). 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.
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| 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 parameterized 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''<sup>2</sup> 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.
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| Geodesics are commonly seen in the study of [[Riemannian geometry]] and more generally [[metric geometry]]. In general relativity, geodesics describe the motion of [[point particle]]s 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.
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| This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s. The article [[geodesic (general relativity)]] discusses the special case of general relativity in greater detail. | |
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| ===Examples===
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| [[File:Transpolar geodesic on a triaxial ellipsoid case A.svg|thumb|right|200px|
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| A [[geodesics on a triaxial ellipsoid|geodesic on a triaxial ellipsoid]].]]
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| The most familiar examples are the straight lines in [[Euclidean geometry]]. On a [[sphere]], the images of geodesics are the [[great circle]]s. The shortest path from point ''A'' to point ''B'' on a sphere is given by the shorter [[arc (geometry)|arc]] of the great circle passing through ''A'' and ''B''. If ''A'' and ''B'' are [[antipodal point]]s (like the [[North Pole]] and the [[South Pole]]), 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).
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| ==Metric geometry==
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| In [[metric geometry]], a geodesic is a curve which is everywhere [[locally]] a [[distance]] minimizer. 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 [[mathematical constant|constant]] ''v'' ≥ 0 such that for any ''t'' ∈ ''I'' there is a neighborhood ''J'' of ''t'' in ''I'' such that for any {{nobr|''t''<sub>1</sub>, ''t''<sub>2</sub> ∈ ''J''}} we have
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| :<math>d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|.\,</math>
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| This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with [[Curve#Length of curves|natural parameterization]], i.e. in the above identity ''v'' = 1 and
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| :<math>d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|.\,</math>
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| If the last equality is satisfied for all ''t''<sub>1</sub>, ''t''<sub>2</sub> ∈''I'', the geodesic is called a '''minimizing geodesic''' or '''shortest path'''.
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| 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 path]]s, although this minimizing sequence need not converge to a geodesic.
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| ==Riemannian geometry==
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| In a [[Riemannian manifold]] ''M'' with [[metric tensor]] ''g'', the length of a continuously differentiable curve γ : [''a'',''b''] → ''M'' is defined by
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| :<math>L(\gamma)=\int_a^b \sqrt{ g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) }\,dt.</math>
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| 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''] → ''M'' such that γ(''a'') = ''p'' and γ(''b'') = ''q''. With this definition of distance, geodesics in a Riemannian manifold are then the locally distance-minimizing paths.
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| The minimizing curves of ''L'' in a small enough [[open set]] of ''M'' can be obtained by techniques of [[calculus of variations]]. Typically, one introduces the following [[action (physics)|action]] or [[energy functional]]
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| :<math>E(\gamma)=\frac{1}{2}\int g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,dt.</math>
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| It is then enough to minimize the functional ''E'', owing to the [[Cauchy–Schwarz inequality]]
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| :<math>L(\gamma)^2\le 2(b-a)E(\gamma)</math>
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| with equality if and only if |dγ/dt| is constant.
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| The [[Euler–Lagrange]] equations of motion for the functional ''E'' are then given in local coordinates by
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| :<math>\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0,</math>
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| where <math>\Gamma^\lambda_{\mu\nu}</math> are the [[Christoffel symbols]] of the metric. This is the '''geodesic equation''', discussed [[#Affine geodesics|below]].
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| ===Calculus of variations===
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| 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
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| :<math>\delta E(\gamma)(\varphi) = \left.\frac{\partial}{\partial t}\right|_{t=0} E(\gamma + t\varphi).</math>
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| The [[critical point (mathematics)|critical point]]s of the first variation are precisely the geodesics. The [[second variation]] is defined by
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| :<math>\delta^2 E(\gamma)(\varphi,\psi) = \left.\frac{\partial^2}{\partial s\partial t}\right|_{s=t=0}E(\gamma + t\varphi + s\psi).</math>
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| In an appropriate sense, zeros of the second variation along a geodesic γ arise along [[Jacobi field]]s. Jacobi fields are thus regarded as variations through geodesics.
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| By applying variational techniques from [[classical mechanics]], one can also regard [[geodesics as Hamiltonian flows]]. They are solutions of the associated [[Hamilton–Jacobi equation]]s, with (pseudo-)Riemannian metric taken as [[Hamiltonian mechanics|Hamiltonian]].
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| ==Affine geodesics==
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| {{Seealso|Geodesics in general relativity}}
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| A '''geodesic''' on a [[Differentiable manifold|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
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| {{NumBlk|:|<math> \nabla_{\dot\gamma} \dot\gamma= 0</math>|{{EquationRef|1}}}}
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| at each point along the curve, where <math>\dot\gamma</math> is the derivative with respect to <math>t</math>. More precisely, in order to define the covariant derivative of <math>\dot\gamma</math> it is necessary first to extend <math>\dot\gamma</math> to a continuously differentiable [[vector field]] in an [[open set]]. However, the resulting value of ({{EquationNote|1}}) is independent of the choice of extension.
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| Using [[local coordinates]] on ''M'', we can write the '''geodesic equation''' (using the [[summation convention]]) as
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| :<math>\frac{d^2\gamma^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{d\gamma^\mu }{dt}\frac{d\gamma^\nu }{dt} = 0\ ,</math>
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| where <math>\gamma^\mu = x^\mu \circ \gamma (t)</math> are the coordinates of the curve γ(''t'') and <math>\Gamma^{\lambda }_{\mu \nu }</math> are the [[Christoffel symbol]]s of the connection ∇. 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 particle]]s in a manifold. Indeed, the equation <math> \nabla_{\dot\gamma} \dot\gamma= 0</math> means that the acceleration 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.
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| ===Existence and uniqueness===
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| 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:
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| :For any point ''p'' in ''M'' and for any vector ''V'' in ''T<sub>p</sub>M'' (the [[tangent space]] to ''M'' at ''p'') there exists a unique geodesic <math>\gamma \,</math> : ''I'' → ''M'' such that
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| ::<math>\gamma(0) = p \,</math> and
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| ::<math>\dot\gamma(0) = V</math>,
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| :where ''I'' is a maximal [[open interval]] in '''R''' containing 0.
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| In general, ''I'' may not be all of '''R''' as for example for an open disc in '''R'''<sup>2</sup>. The proof of this theorem follows from the theory of [[ordinary differential equation]]s, 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|smoothly]] on both ''p'' and ''V''.
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| ===Geodesic flow===
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| Geodesic [[Flow (mathematics)|flow]] is a local '''R'''-[[group action|action]] on [[tangent bundle]] ''TM'' of a manifold ''M'' defined in the following way
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| :<math>G^t(V)=\dot\gamma_V(t)</math> | |
| where ''t'' ∈ '''R''', ''V'' ∈ ''TM'' and <math>\gamma_V</math> denotes the geodesic with initial data <math>\dot\gamma_V(0)=V</math>. Thus, ''G<sub>t</sub>''(''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''.
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| On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a [[Hamiltonian flow]] on the cotangent bundle. The [[Hamiltonian mechanics|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 <math>g</math>, i.e.
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| :<math>g(G^t(V),G^t(V))=g(V,V)</math>.
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| In particular, when ''V'' is a unit vector, <math>\gamma_V</math> remains unit speed throughout, so the geodesic flow is tangent to the [[unit tangent bundle]]. [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] implies invariance of a kinematic measure on the unit tangent bundle.
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| ===Geodesic spray===
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| 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 [[spray (mathematics)|geodesic spray]].
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| More precisely, an affine connection gives rise to a splitting of the [[double tangent bundle]] TT''M'' into [[horizontal bundle|horizontal]] and [[vertical bundle]]s:
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| :<math>TTM = H\oplus V.</math>
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| The geodesic spray is the unique horizontal vector field ''W'' satisfying
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| :<math>\pi_* W_v = v\,</math>
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| at each point ''v'' ∈ T''M''; here π<sub>∗</sub> : TT''M'' → T''M'' denotes the [[pushforward (differential)]] along the projection π : T''M'' → ''M'' associated to the tangent bundle.
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| 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 T''M'' \ {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
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| :<math>H_{\lambda X} = d(S_\lambda)_X H_X\,</math>
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| for every ''X'' ∈ T''M'' \ {0} and λ > 0. Here ''d''(''S''<sub>λ</sub>) is the [[pushforward (differential)|pushforward]] along the scalar homothety <math>S_\lambda: X\mapsto \lambda X.</math> A particular case of a non-linear connection arising in this manner is that associated to a [[Finsler manifold]].
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| ===Affine and projective geodesics===
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| Equation ({{EquationNote|1}}) is invariant under affine reparameterizations; that is, parameterizations of the form
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| :<math>t\mapsto at+b</math>
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| 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'''.
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| An affine connection is ''determined by'' its family of affinely parameterized geodesics, up to [[torsion tensor|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 <math>\nabla, \bar{\nabla}</math> are two connections such that the difference tensor
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| :<math>D(X,Y) = \nabla_XY-\bar{\nabla}_XY</math>
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| is [[skew-symmetric]], then <math>\nabla</math> and <math>\bar{\nabla}</math> have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as <math>\nabla</math>, but with vanishing torsion.
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| Geodesics without a particular parameterization are described by a [[projective connection]].
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2;"> | |
| * [[Basic introduction to the mathematics of curved spacetime]]
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| * [[Clairaut's relation]]
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| * [[Closed geodesic]]
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| * [[Complex geodesic]]
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| * [[Differential geometry of curves]]
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| * [[Exponential map]]
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| * [[Fermat's principle]]
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| * [[Geodesic dome]]
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| * [[Geodesic (general relativity)]]
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| * [[Geodesics as Hamiltonian flows]]
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| * [[Hopf–Rinow theorem]]
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| * [[Intrinsic metric]]
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| * [[Jacobi field]]
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| * [[Quasigeodesic]]
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| * [[Solving the geodesic equations]]
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| * [[Zoll surface]]
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| * [[Nautical chart]]
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| * [[Rhumb line]] (loxodrome)
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| * [[Meridian arc]]
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| </div> | |
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| ==References==
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| *{{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''.
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| *{{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham | 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''.
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| *{{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''.
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| *{{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}}.
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| *{{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''.
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| *{{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 (book)|Gravitation]] | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973}}
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| *{{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.
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| *{{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}}
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| *{{springer|first=Yu.A.|last=Volkov|title=Geodesic line|id=G/g044120}}.
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| *{{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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| == External links ==
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| * [http://www.black-holes.org/relativity5.html Caltech Tutorial on Relativity] — A nice, simple explanation of geodesics with accompanying animation.
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| * [http://www.cmsim.eu/papers_pdf/january_2012_papers/25_CMSIM_2012_Pokorny_1_281-298.pdf Geodesics Revisited] — Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on a [[torus]]), mechanics ([[brachistochrone]]) and optics (light beam in inhomogeneous medium).
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| * [http://wiki.sagemath.org/interact/geometry#Geodesics_on_a_parametric_surface Geodesics on a parametric surface -- sage interact] — Interactive [[Sagemath]] worksheet to calculate and illustrate geodesics on parametric surfaces.
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| * [http://www.map.mpim-bonn.mpg.de/Totally_geodesic_submanifold Totally geodesic submanifold] at the Manifold Atlas
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| {{tensor}}
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| [[Category:Geodesic (mathematics)| ]]
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