We explain a general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. There is an evidence that the described results are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory.
