Comparative analysis for pair of dynamical systems, one of which is   Lagrangian
  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. 
Ruslan Sharipov