An Expression of Classical Dynamics

Abstract

The proposed formulation extends the Euler variable approach, classical in Continuum Mechanics, up to make it valid for such singular systems as, for instance, a single mass-point. The key concept is the kinetic tensor measure of the investigated material, relative to some window in time-space. This is first developed in the framework of Galilean time-space. In that case, the fundamental equation involves the four-dimensional vector distribution divergence of the kinetic tensor measure. It is shown, in particular, how the initial conditions of an evolution problem or the confinement of the investigated system by a given boundary, possibly with shocks, may be described through adequate terms in the fundamental equation. In order to develop similar procedures in the Riemannian manifold setting of Analytical Dynamics, one introduces the differential operator equilibrium, acting on the doubly contravariant symmetric tensor measures of the manifold. This operator receives a variational interpretation, in terms of the transport by test flows. Thereby, the connection of the proposed formulation of Dynamics with Hamilton’s principle is explained.