მანძილი: განსხვავება გადახედვებს შორის

</center>

Here <math>\vec{r}(t)</math> is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the minimal value of this integral and is obtained when <math>r = r^{*}</math> where <math>r^{*}</math> is the optimal trajectory. In the familiar Euclidean case (the above integral) this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the [[Euler-Lagrange equations]] for the above [[functional (mathematics)|functional]]. In [[non-Euclidean geometry|non-Euclidean]] manifolds (curved spaces) where the nature of the space is represented by a [[metric (mathematics)|metric]] <math>g_{ab}</math> the integrand has be to modified to <math>\sqrt{g^{ac}\dot{r}_{c}g_{ab}\dot{r}^{b}}</math>, where [[Einstein summation convention]] has been used.

 

===General case===