Some Singular Relativistic Properties of the One-Dimensional Models

The so-called no-interaction Theorem of D.G.Currie, T.F.Jordan, E.C.Sudarshan, H.Leutwyler, G.Marmo and N.Mukunda makes it possible to construct relativistic quasi-classical particle dynamics only in the post-galilean approximation. We call post-galilean the approximation in which the corrections of order O(c^-2) ( c denoting the light speed) to the Galilei group are taken into account. We found that in this approximation the Lagrangians are singular on some surfaces of the phase space. These peculiarities have different physical and mathematical nature from the ones studied by P.M.A.Dirac, where Hessians vanish in the whole phase-space. The dynamic properties develop peculiar characteristics on the singular surfaces. Similar singular surfaces appear in the post-newtonian approximation, when the corrections of order O(c^-2) to the gravitational theory of Newton occur. As a consequence of the dynamic properties on the singular mani-folds we can give a new interpretation of some well known physical phenomena, such as the so-called 'radius of electron', the 'radius of Schwartzschild', the strongly connected character of the electron-positron pair and the irreversibility of motion in classical mechanics (with the consequence the relaxation to the equilibrium distribution). This paper is concerned only with the study the one-dimensional models. Two and three-dimensional systems will be examined in a later publication.