Geodesic

The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations. This has some minor technical problems, because there is an infinite-dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f(s) to f(t) along the curve equals |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 between two points on a sphere is a geodesic but not the shortest path between the points. The map ${\displaystyle t\to t^{2}}$ 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 and more generally metric geometry. In general relativity, geodesics in spacetime describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved spacetime. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian manifolds. The article Levi-Civita connection discusses the more general case of a pseudo-Riemannian manifold and geodesic (general relativity) discusses the special case of general relativity in greater detail.

Examples

A geodesic on a triaxial ellipsoid.

If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.

The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points, then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

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 constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t₁, t₂ ∈ J we have

${\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=v\left|t_{1}-t_{2}\right|.}$

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization, i.e. in the above identity v = 1 and

${\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|.}$

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.

In a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ : [a,b] → M is defined by

${\displaystyle L(\gamma )=\int _{a}^{b}{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt.}$

The distance d(p, q) between two points p and q of M is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ : [a,b] → M such that γ(a) = p and γ(b) = q. In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the following action or energy functional

${\displaystyle E(\gamma )={\frac {1}{2}}\int _{a}^{b}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt.}$

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 ${\displaystyle C^{1}}$ curve (more generally, a ${\displaystyle W^{1,2}}$ curve), the Cauchy–Schwarz inequality gives

${\displaystyle L(\gamma )^{2}\leq 2(b-a)E(\gamma )}$

with equality if and only if ${\displaystyle g(\gamma ',\gamma ')}$ is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of ${\displaystyle E(\gamma )}$ also minimize ${\displaystyle L(\gamma )}$, 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 ${\displaystyle \gamma }$, 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 ${\displaystyle L(\gamma )}$ are generally not very regular, because arbitrary reparameterizations are allowed.

The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by

${\displaystyle {\frac {d^{2}x^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}=0,}$

where ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$ are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.

A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so

${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}$

(1)

at each point along the curve, where ${\displaystyle {\dot {\gamma }}}$ is the derivative with respect to ${\displaystyle t}$. More precisely, in order to define the covariant derivative of ${\displaystyle {\dot {\gamma }}}$ it is necessary first to extend ${\displaystyle {\dot {\gamma }}}$ to a continuously differentiable vector field in an open set. However, the resulting value of (1) is independent of the choice of extension.

Using local coordinates on M, we can write the geodesic equation (using the summation convention) as

${\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ ,}$

where ${\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}$ are the coordinates of the curve γ(t) and ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$ are the Christoffel symbols of the connection ∇. This is an ordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold. Indeed, the equation ${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}$ 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.

Efficient solvers for the minimal geodesic problem on surfaces posed as eikonal equations have been proposed by Kimmel and others.[3][4]

Geodesics serve as the basis to calculate:

• geodesic airframes; see geodesic airframe or geodetic airframe
• geodesic structures – for example geodesic domes
• horizontal distances on or near Earth; see Earth geodesics
• mapping images on surfaces, for rendering; see UV mapping
• particle motion in molecular dynamics (MD) computer simulations[5]
• robot motion planning (e.g., when painting car parts); see Shortest path problem

• Spivak, Michael (1999), A Comprehensive introduction to differential geometry (Volume 2), Houston, TX: Publish or Perish, ISBN 978-0-914098-71-3

Wikimedia Commons has media related to Geodesic (mathematics).

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• 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).
• Geodesics on a parametric surface -- sage interact — Interactive SageMath worksheet to calculate and illustrate geodesics on parametric surfaces.
• Totally geodesic submanifold at the Manifold Atlas