On a variational property of geodesics in Riemannian and Finsler spaces

A new varionational property of geodesics in (pseudo-) Riemannian and Finsler spaces has been found. It is proved the only geodesics $x=x(t)$ with the canonical parameter are stacionary with respect to the integral $\int^{t_1}_{t_0}f(g_{ij}(x,\dot{x})\dot{x}^i\dot{x}^j)dt$, where $f$ ($f'\neq 0$) is a square free two differentiable function, $g_{ij}$ are components of the metric tensor, and $\dot{x}(t)=dx(t)/dt$.