  It is known that some equations of differential geometry are derived from
variational principle in form of Euler-Lagrange equations. The equations of
geodesic flow in Riemannian geometry is an example. Conversely, having
Lagrangian dynamical system in a manifold, one can consider it as geometric
equipment of this manifold. Then properties of other dynamical systems can be
studied relatively as compared to this Lagrangian one. This gives fruitful
analogies for generalization. In present paper theory of normal shift of
hypersurfaces is generalized from Riemannian geometry to the geometry
determined by Lagrangian dynamical system. Both weak and additional normality
equations for this case are derived.
