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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
" to "curved space
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s". In presence of a metric
Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection
Affine connection

In the mathematics of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits vector field to be derivative as if they were functions on the manifold with values in a fixed vector space....
, geodesics are defined to be curves whose tangent vector
Tangent vector

A tangent vector is a Vector that follows the direction of a curve or a surface at a given point.* Differential geometry of curves, description in the context of curves in Rn....
s remain parallel if they are transported
Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection....
 along it.

The term "geodesic" comes from geodesy
Geodesy

Geodesy , also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space....
, the science of measuring the size and shape of Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
; in the original sense, a geodesic was the shortest route between two points on the Earth's surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, namely, a segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
 of a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
" to "curved space
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s". In presence of a metric
Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection
Affine connection

In the mathematics of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits vector field to be derivative as if they were functions on the manifold with values in a fixed vector space....
, geodesics are defined to be curves whose tangent vector
Tangent vector

A tangent vector is a Vector that follows the direction of a curve or a surface at a given point.* Differential geometry of curves, description in the context of curves in Rn....
s remain parallel if they are transported
Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection....
 along it.

The term "geodesic" comes from geodesy
Geodesy

Geodesy , also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space....
, the science of measuring the size and shape of Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
; in the original sense, a geodesic was the shortest route between two points on the Earth's surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, namely, a segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
 of a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory
Graph theory

In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
, one might consider a geodesic between two vertices/nodes of a graph.

Geodesics are of particular importance in general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, as they describe the motion of inertial test particles.

Introduction

The shortest path between two points in a curved space can be found by writing the equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
 for the length of a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations
Calculus of variations

Calculus of variations is a field of mathematics that deals with functional , as opposed to ordinary calculus which deals with function . Such functionals can for example be formed as integrals involving an unknown function and its derivatives....
. This has some minor technical problems, because there is an infinite dimensional space of different ways to parametrize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parametrized "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
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 of the curve; minimizing the energy leads to the same equations for a "constant velocity" 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.

In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parametrized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t?t2 from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study of Riemannian geometry
Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, manifold with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smooth function from point to point....
 and more generally metric geometry. In relativistic physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, geodesics describe the motion of point particle
Point particle

A point particle is an idealized object heavily used in physics. Its defining feature is that it lacks dimension extension: being zero-dimensional, it does not take up space....
s under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite
Satellite

In the context of spaceflight, a satellite is an Physical body which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
, or the shape of a planetary orbit
Planetary orbit

In physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
 are all geodesics in curved space-time. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
 and pseudo-Riemannian manifold
Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann....
s. The article geodesic (general relativity)
Geodesic (general relativity)

In general relativity, geodesics generalize the notion of "straight lines" to curved spacetime. This concept is based on the mathematical concept of a geodesic....
 discusses the special case of general relativity in greater detail.

Examples

The most familiar examples are the straight lines in Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
. On a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, the images of geodesics are the great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
s. 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 point
Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diameter opposite it ? so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
s (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
Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
 minimizer. More precisely, a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 ?: I ? M from an interval I of the reals to the metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 M is a geodesic if there is a constant
Mathematical constant

A mathematical constant is a number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement....
 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

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parametrization
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
, i.e. in the above identity v = 1 and

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. 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.

(Pseudo-)Riemannian geometry

A geodesic on a smooth manifold M with an affine connection
Affine connection

In the mathematics of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits vector field to be derivative as if they were functions on the manifold with values in a fixed vector space....
  ? is defined as a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 ?(t) such that parallel transport
Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection....
 along the curve preserves the tangent vector to the curve, so at each point along the curve, where is the derivative with respect to .

Using local coordinates
Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc....
 on M, we can write the geodesic equation (using the summation convention) as where are the coordinates of the curve ?(t) and are the Christoffel symbols of the connection ?. This is just an ordinary differential equation
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
 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
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
, geodesics can be thought of as trajectories of free particle
Free particle

In physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "field-free" space....
s in a manifold.Indeed, the equation 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 determinded by the bending of the surface. This is also the idea of the general relativity where particles move on geodesics and the bending is caused by the gravity.

Geodesics for a (pseudo
Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann....
-)Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
 M are defined to be geodesics for its Levi-Civita connection
Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is the Torsion -free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric....
. In a Riemannian manifold a geodesic is the same as a curve that locally minimizes the length and is parametrized so that the tangent vector has constant length. Geodesics can also be defined as extremal
Stationary point

In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
 curves for the following action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
  energy functional where 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
Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
. See Riemannian manifolds in Hamiltonian mechanics
Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
 for further details.

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; this is a variant of the Frobenius theorem
Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations....
. More precisely:

For any point p in M and for any vector V in TpM (the tangent space
Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
 to M at p) there exists a unique geodesic : I → M such that
and
,
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 equation
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
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. ? depends smoothly
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
 on both p and V.

Geodesic flow

Geodesic flow
Flow (mathematics)

In mathematics, a flow formalizes, in mathematical terms, the general idea of "a variable that depends on time" that occurs very frequently in engineering, physics and the study of ordinary differential equations....
 is an -action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 on tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
  of a manifold defined in the following way where , and denotes the geodesic with initial data .

It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian
Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* Hamiltonian ...
. In particular it preserves the (pseudo-)Riemannian metric , i.e. . That makes possible to define geodesic flow on unit tangent bundle
Unit tangent bundle

In mathematics, the unit tangent bundle of a Finsler manifold , denoted by UT or simply UTM, is a fiber bundle over M given by the disjoint union...
  of the Riemannian manifold when the geodesic is of unit speed.

A closed orbit of the geodesic flow corresponds to a closed geodesic
Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M....
 on .

Geodesic spray

The geodesic flow defines a family of curves in the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
. The derivatives of these curves define a vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
  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
Affine connection

In the mathematics of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits vector field to be derivative as if they were functions on the manifold with values in a fixed vector space....
. A curve in such a manifold is a geodesic if its tangent vector remains parallel to the curve when it is transported along it. Geodesics defined in this way carry a preferred class of affine parametrizations. These are those parametrizations for which

This equation is invariant under affine reparametrizations; that is, parametrizations of the form where a and b are constant real numbers.

An affine connection is determined by its family of affinely parameterized geodesics, up to torsion
Torsion tensor

In differential geometry, the notion of 'torsion' is a manner of characterizing a twist or screw theory of a moving frame around a curve. The torsion of curves, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the geometry of surfaces, the geodesic...
 . 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 are two connections such that the difference tensor is skew-symmetric, then and have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as , but with vanishing torsion.

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

In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection....
.

See also


  • Zoll surface
    Zoll surface

    A Zoll surface, named after Otto Zoll, , is a surface homeomorphic to the 2-sphere, with a Riemannian metric all of whose geodesics are closed geodesic and of equal length....


External links

  • — A nice, simple explanation of geodesics with accompanying animation.