Editing Geodesic (section)

===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]].