From integral manifolds and metrics to potential maps

Abstract

Our paper contains two main results: (1) the integral manifolds of a distribution together with two Riemann metrics produce potential maps which are in fact least squares approximations of the starting integral manifolds; (2) the least squares energy admits extremals satisfying periodic boundary conditions. Section 1 contains historical and bibliographical notes. Section 2 analyses some elements of the geometry produced on the jet bundle of order one by a semi-Riemann Sasaki-like metric. Section 3 describes the maximal integral manifolds of a distribution as solutions of a PDEs system of order one. Section 4 studies Poisson-like second-order prolongations of first order PDE systems and formulates the Lorentz-Udriste World-Force Law on a suitable semi-Riemann-Lagrange manifold (the base manifold of the jet bundle of order one). Section 5 exploits the idea of least squares Lagrangians, to include the integral manifolds of a distribution into a class of extremals. Section 6 gives conditions for the existence of extremals in conditions of multi-periodicity. Section 7 refers to the canonical forms of the vertical metric d-tensor produced by a density of energy on jet bundle of order one.