Newtonian normal shift in multidimensional Riemannian geometry (Q2784548)

In the setting of the revision of the main results of his PhD thesis, the author presents an explicit description of Newtonian dynamical systems admitting a normal shift in Riemannian manifolds of dimension bigger or equal than three. These systems constitute a generalization of the geodesic normal shift of a hypersurface \(S\) in a Riemannian manifold \(M\). The geodesic normal shift of hypersurfaces is constructed by moving the points \(p\) in the hypersurface \(S\) along the normal geodesics to \(S\) starting at these points. These normal geodesics, called the shift trajectories are described, in local coordinates, by the classical system of homogeneous ordinary differential equations involving the Christoffel symbols of the connection. Then, each point \(p\) in \(S\) is joined with the point of the geodesic \(p_t\), so that we obtain for each value of the parameter \(t\) an application \(f_t\) of \(S\) onto its image \(S_t\), in such a way that it can be proved that all hypersurfaces \(S_t\) are orthogonal to the shift trajectories.NEWLINENEWLINENEWLINEThe generalization proposed by the author consists in replacing the zero right-side in the geodesic equations in local coordinates by the components of a function \(F\) depending on the local coordinates of the curve and its first derivatives. When the Riemannian manifold is the Euclidean three-space and the Christoffel symbols are equal to zero, these new equations express Newton's Second Law and describe the motion of a material point of unit mass in the forced field determined by \(F\). In the case of an arbitrary Riemannian manifold, they describe the dynamics of complex mechanical systems with holonomic constraints.NEWLINENEWLINENEWLINEThe proposed construction, called in analogy to Newton's construction ``Newtonian dynamical system on the Riemannian manifold \(M\), with force field \(F\)'', is local in the sense that the shift map \(f_t\) can be defined in this setting in the same way as in the case of a usual geodesic shift. In this case it is well defined in a neighbourhood of each point of the hypersurface \(S\), and for values of \(t\) nearby of zero it is guaranteed that the trajectories of a Newtonian dynamical system are orthogonal to the hypersurface \(S\) at \(t=0\), but not for all \(t\), (as in the case of the geodesic normal shift).NEWLINENEWLINENEWLINEIn the paper the factors governing the normal property of the Newtonian dynamical systems on a Riemannian manifold are studied, giving a description of this normal property, defined in a local way in terms of some equations satisfied by the force field \(F\) and called the ``normality equations''. The analysis of these normality equations lead the author to the statement of a formula defining the force field \(F\) of an arbitrary Newtonian dynamical system admitting normal shift, (in the terms established in the paper) on a Riemannian manifold of dimension bigger than or equal to three. This formula allows to give a detailed description of the kinematics of the normal shifts of this type.