On the Concept of Normal Shift in Non-metric Geometry

Theory of Newtonian dynamical systems admitting normal shift of hypersurfaces was first developed for the case of Riemannian manifolds. Recently it was generalized for manifolds geometric equipment of which is given by some regular Lagrangian or, equivalently, by some regular Hamiltonian dynamical system. In present paper we consider further generalization of this theory for the case, when geometry of manifold is given by generalized Legendre transformation. 1. What is normal shift ? Brief historical overview. Phenomenon of normal shift is very simple by its nature. Let’s consider it in three-dimensional Euclidean space R. Suppose that σ is some smooth orientable surface in R. At each point p ∈ σ one can draw unit normal vector n such that n = n(p) would be a smooth vector-valued function on σ. Let’s move each point p of σ in the direction of vector n(p) to the distance t which is the same for all points p ∈ σ. Then moved points pt would form another surface σt as shown on Fig. 1.1. Changing parameter t we would obtain one-parametric family of surfaces. This construction is known as Bonnet transformation. In Bonnet construction initial surface σ is transformed by moving each point of σ. Trajectories of motion in this case are straight lines directed along normal vectors and points of σ move along them with a constant speed |v| = 1. Therefore parameter t, which is the distance of displacement, can also be interpreted as time variable. Bonnet noted that all surfaces σt in his construction are perpendicular to the trajectories of moving points. For this reason his construction is also known as normal displacement or normal shift. Basic observation by Bonnet, i. e. orthogonality of surfaces σt and shift trajectories, gave an impetus for generalization of his construction. This was done by me 1991 Mathematics Subject Classification. 53D20, 70G45.