Editing Geodesic (section)

==Riemannian geometry==
In a [[Riemannian manifold]] <math>M</math> with [[metric tensor]] <math>g</math>, the length <math>L</math> of a continuously differentiable curve <math>\gamma : [a,b] \to M</math> is defined by
:<math>L(\gamma)=\int_a^b \sqrt{  g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) }\,dt.</math>
The distance <math>d(p,q)</math> between two points <math>p</math> and <math>q</math> of <math>M</math> is defined as the [[infimum]] of the length taken over all continuous, piecewise continuously differentiable curves <math>\gamma : [a,b]\to M</math> such that <math>\gamma(a)=p</math> and <math>\gamma(b)=q</math>. 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 (physics)|action]] or [[energy functional]]
:<math>E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,dt.</math>
All minima of <math>E</math> are also minima of <math>L</math>, but <math>L</math> is a bigger set since paths that are minima of <math>L</math> can be arbitrarily re-parameterized (without changing their length), while minima of <math>E</math> cannot.
For a piecewise <math>C^1</math> curve (more generally, a <math>W^{1,2}</math> curve), the [[Cauchy–Schwarz inequality]] gives
:<math>L(\gamma)^2 \le 2(b-a)E(\gamma)</math>
with equality if and only if <math>g(\gamma',\gamma')</math> is equal to a constant a.e.; the path should be travelled at constant speed.  It happens that minimizers of <math>E(\gamma)</math> also minimize <math>L(\gamma)</math>, 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 <math>E</math> is a more robust variational problem.  Indeed, <math>E(\gamma)</math> is a "convex function" of <math>\gamma</math>, 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 <math>L(\gamma)</math> are generally not very regular, because arbitrary reparameterizations are allowed.

The [[Euler–Lagrange equation]]s of motion for the functional <math>E</math> are then given in local coordinates by
:<math>\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0,</math>
where <math>\Gamma^\lambda_{\mu\nu}</math> are the [[Christoffel symbols]] of the metric.  This is the '''geodesic equation''', discussed [[#Affine geodesics|below]].

===Calculus of variations===

Techniques of the classical [[calculus of variations]] can be applied to examine the energy functional <math>E</math>.  The [[first variation]] of energy is defined in local coordinates by

:<math>\delta E(\gamma)(\varphi) = \left.\frac{\partial}{\partial t}\right|_{t=0} E(\gamma + t\varphi).</math>

The [[critical point (mathematics)|critical point]]s of the first variation are precisely the geodesics.  The [[second variation]] is defined by

:<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>

In an appropriate sense, zeros of the second variation along a geodesic <math>\gamma</math> arise along [[Jacobi field]]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 equation]]s, with [[Pseudo Riemannian metric|(pseudo-)Riemannian metric]] taken as [[Hamiltonian mechanics|Hamiltonian]].