I know that the critical points of the energy functional
$$E(\gamma) = \frac{1}{2}\int_a^b |\gamma'(t)|^2\, dt$$
are geodesics. As finding geodesics sometimes involves us with some complicated system of equations, this trick is useful to find the critical points of $E$.
Can someone give some example to see how it works in the Riemannian case?
A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + \sum _{i,j\geq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction. This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $\gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $\int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..