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In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, a geodesic () is commonly a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
representing in some sense the shortest path ( arc) between two points in a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
, or more generally in a
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
. The term also has meaning in any
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
with a connection. It is a generalization of the notion of a "
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...
" to a more general setting. The noun ''geodesic'' and the adjective ''geodetic'' come from ''
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
'', the science of measuring the size and shape of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

Earth
, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
. For a
spherical Earth Spherical Earth or Earth's curvature refers to the approximation of as a . The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of . In the 3rd century BC, established ...
, it is a
segment Segment or segmentation may refer to: Biology *Segmentation (biology), the division of body plans into a series of repetitive segments **Segmentation in the human nervous system *Internodal segment, the portion of a nerve fiber between two Nodes of ...

segment
of a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
(see also
great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the shortest distance between two points on the surface of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical ob ...
). The term has been generalized to include measurements in much more general mathematical spaces; for example, in
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, one might consider a
geodesic In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
between two vertices/nodes of a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing
geodesic curvatureIn Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
. More generally, in the presence of an
affine connection In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
, a geodesic is defined to be a curve whose
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
s remain parallel if they are
transported ''Transported'' is an Australian convict melodrama film directed by W. J. Lincoln. It is considered a lost film. Plot In England, Jessie Grey is about to marry Leonard Lincoln but the evil Harold Hawk tries to force her to marry him and she woun ...

transported
along it. Applying this to the
Levi-Civita connection In Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo Riemannian geometry (in particular the Lorentzian manifold, Lorentzian geometry of General Relativity, general relativity), the Levi-Civita connection is the unique affine co ...
of a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...
recovers the previous notion. Geodesics are of particular importance in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. Timelike
geodesics in general relativity In general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the o ...
describe the motion of
free fall #REDIRECT Free fall #REDIRECT Free fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—inc ...

free fall
ing
test particlesIn Theoretical physics, physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge (physics), charge, or volume, size) are assumed to be negligible except for the property be ...
.


Introduction

A locally shortest path between two given points in a curved space, assumed to be a
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
, can be defined by using the
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
for the
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
(a function ''f'' from an
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of
R
R
to the space), and then minimizing this length between the points using the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathemati ...
. 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 , ''s''−''t'', . 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). 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 A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
between two points on a sphere is a geodesic but not the shortest path between the points. The map t \to t^2 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#REDIRECT Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...
and more generally
metric geometry In mathematics, a metric space is a non empty Set (mathematics), set together with a Metric (mathematics)#Definition, metric on the set. The metric is a function (mathematics), function that defines a concept of ''distance'' between any two Elemen ...
. In
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, geodesics in
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
describe the motion of
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, ...
s under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting
satellite In the context of spaceflight Spaceflight (or space flight) is an application of astronautics to fly spacecraft into or through outer space, either human spaceflight, with or uncrewed spaceflight, without humans on board. Most spaceflight ...

satellite
, or the shape of a
planetary orbit Planetary means relating to a planet A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough ...
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 manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
s. The article
Levi-Civita connection In Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo Riemannian geometry (in particular the Lorentzian manifold, Lorentzian geometry of General Relativity, general relativity), the Levi-Civita connection is the unique affine co ...
discusses the more general case of a
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...
and
geodesic (general relativity) In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freel ...
discusses the special case of general relativity in greater detail.


Examples

The most familiar examples are the straight lines in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
. On a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
, the images of geodesics are the
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
s. 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 point In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, then there are ''infinitely many'' shortest paths between them.
Geodesics on an ellipsoid The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodesic' ...
behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).


Triangles

A geodesic triangle is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics are
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
arcs, forming a
spherical triangle Spherical trigonometry is the branch of spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (plan ...

spherical triangle
.


Metric geometry

In
metric geometry In mathematics, a metric space is a non empty Set (mathematics), set together with a Metric (mathematics)#Definition, metric on the set. The metric is a function (mathematics), function that defines a concept of ''distance'' between any two Elemen ...
, a geodesic is a curve which is everywhere
locallyIn mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighbourhood (mathematics), ne ...
a
distance Distance is a numerical measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and ...

distance
minimizer. More precisely, a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
from an interval ''I'' of the reals to the
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
''M'' is a geodesic if there is a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
such that for any there is a neighborhood ''J'' of ''t'' in ''I'' such that for any we have :d(\gamma(t_1),\gamma(t_2)) = v \left, 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 parameterization, i.e. in the above identity ''v'' = 1 and :d(\gamma(t_1),\gamma(t_2)) = \left, t_1 - t_2 \ . If the last equality is satisfied for all , 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 In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
''M'' with
metric tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
''g'', the length ''L'' of a continuously differentiable curve γ :  'a'',''b''nbsp;→ ''M'' is defined by :L(\gamma)=\int_a^b \sqrt\,dt. The distance ''d''(''p'', ''q'') between two points ''p'' and ''q'' of ''M'' is defined as the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
of the length taken over all continuous, piecewise continuously differentiable curves γ :  'a'',''b''nbsp;→ ''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 ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
or
energy functional The energy functional is the total energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its ...
:E(\gamma)=\frac\int_a^b g_(\dot\gamma(t),\dot\gamma(t))\,dt. 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 (without changing their length), while minima of ''E'' cannot. For a piecewise C^1 curve (more generally, a W^ curve), the
Cauchy–Schwarz inequality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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.; the path should be travelled at constant speed. 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 equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
s of motion for the functional ''E'' are then given in local coordinates by :\frac + \Gamma^_\frac\frac = 0, where \Gamma^\lambda_ are the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metric ...
of the metric. This is the geodesic equation, discussed
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
.


Calculus of variations

Techniques of the classical
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathemati ...
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\_ E(\gamma + t\varphi). The critical points of the first variation are precisely the geodesics. The
second variation The second (symbol: s, abbreviation: sec) is the SI base unit, base unit of time in the International System of Units (SI) (French: Système International d’unités), commonly understood and historically defined as of a day – this factor d ...
is defined by :\delta^2 E(\gamma)(\varphi,\psi) = \left.\frac \_ E(\gamma + t\varphi + s\psi). In an appropriate sense, zeros of the second variation along a geodesic γ arise along
Jacobi fieldIn Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form ...
s. 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 equationHamilton may refer to: * Alexander Hamilton (1755–1804), first American Secretary of the Treasury and one of the Founding Fathers of the United States **Hamilton (musical), ''Hamilton'' (musical), a 2015 Broadway musical written by Lin-Manuel Mira ...
s, with (pseudo-)Riemannian metric taken as Hamiltonian.


Affine geodesics

A geodesic on a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
''M'' with an
affine connection In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
∇ is defined as a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
γ(''t'') such that
parallel transport In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

parallel transport
along the curve preserves the tangent vector to the curve, so at each point along the curve, where \dot\gamma is the derivative with respect to t. More precisely, in order to define the covariant derivative of \dot\gamma it is necessary first to extend \dot\gamma to a continuously differentiable
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
in an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. However, the resulting value of () is independent of the choice of extension. Using
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow t ...
on ''M'', we can write the geodesic equation (using the
summation convention In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) as :\frac + \Gamma^_\frac\frac = 0\ , where \gamma^\mu = x^\mu \circ \gamma (t) are the coordinates of the curve γ(''t'') and \Gamma^_ are the
Christoffel symbol In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s of the connection ∇. This is an
ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
s in a manifold. Indeed, the equation \nabla_ \dot\gamma= 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 gravity.


Existence and uniqueness

The ''local existence and uniqueness theorem'' for geodesics states that geodesics on a smooth manifold with an
affine connection In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
exist, and are unique. More precisely: :For any point ''p'' in ''M'' and for any vector ''V'' in ''TpM'' (the
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
in R containing 0. The proof of this theorem follows from the theory of
ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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
smoothly
smoothly
on both ''p'' and ''V''. In general, ''I'' may not be all of R as for example for an open disc in R2. Any extends to all of if and only if is geodesically complete.


Geodesic flow

Geodesic
flow Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on ...
is a local R-
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
on the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

tangent bundle
''TM'' of a manifold ''M'' defined in the following way :G^t(V)=\gamma_V(t) where ''t'' ∈ R, ''V'' ∈ ''TM'' and \gamma_V denotes the geodesic with initial data \dot\gamma_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 geodesicIn differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic, geo ...
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-formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. 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 Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

tangent bundle
. The derivatives of these curves define a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
on the
total space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
of the tangent bundle, known as the geodesic spray. More precisely, an affine connection gives rise to a splitting of the double tangent bundle TT''M'' into horizontal and
vertical bundle In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle. The vertical bundle consists of all vectors that are ...
s: :TTM = H\oplus V. The geodesic spray is the unique horizontal vector field ''W'' satisfying :\pi_* W_v = v\, at each point ''v'' ∈ T''M''; here π : TT''M'' → T''M'' denotes the
pushforward (differential) In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differen ...
along the projection π : T''M'' → ''M'' associated to the tangent bundle. More generally, the same construction allows one to construct a vector field for any
Ehresmann connection In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...

Ehresmann connection
on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle T''M'' \ ) 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_ = d(S_\lambda)_X H_X\, for every ''X'' ∈ T''M'' \  and λ > 0. Here ''d''(''S''λ) is the
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" ope ...
along the scalar homothety S_\lambda: X\mapsto \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 () is invariant under affine reparameterizations; that is, parameterizations of the form :t\mapsto 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 () are called geodesics with affine parameter. 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 are two connections such that the difference tensor :D(X,Y) = \nabla_XY-\bar_XY is skew-symmetric, then \nabla and \bar 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 connectionIn differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...
.


Computational methods

Efficient solvers for the minimal geodesic problem on surfaces posed as eikonal equations have been proposed by Kimmel and others.


Ribbon Test

A Ribbon "Test" is a way of finding a geodesic on a 3-dimensional curved shape. When a ribbon is wound around a cone, a part of the ribbon does not touch the cone's surface. If the ribbon is wound around a different curved path, and all the particles in the ribbon touch the cone's surface, the path is a ''Geodesic''.


Applications

Geodesics serve as the basis to calculate: * geodesic airframes; see
geodesic airframe A geodetic airframe is a type of construction for the airframes of aircraft developed by United Kingdom, British aeronautical engineer Barnes Wallis in the 1930s (who sometimes spelled it "geodesic"). Earlier, it was used by Prof. Schütte for the ...
or
geodetic airframe A geodetic airframe is a type of construction for the airframe The mechanical structure of an aircraft is known as the airframe. This structure is typically considered to include the fuselage, Landing gear, undercarriage, empennage and wings, a ...
* geodesic structures – for example
geodesic domes A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress (physics), stress throughout the structure, mak ...
* horizontal distances on or near Earth; see Earth geodesics * mapping images on surfaces, for rendering; see UV mapping * particle motion in Molecular dynamics, molecular dynamics (MD) computer simulations * robot motion planning (e.g., when painting car parts); see Shortest path problem


See also

* * * * Differential geometry of surfaces * Geodesic circle * * * * * * *


Notes


References

*


Further reading

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


External links


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).
Totally geodesic submanifold
at the Manifold Atlas {{Authority control Differential geometry Geodesic (mathematics),