Forces in Newtonian dynamics and forces in Eulerian dynamics

Abstract

The essential distinction between the Newtonian mass-point dynamics and th Eulerian rigid body dynamics consist in the fact that while the former is based upon, and is developed from, a single fundamental dynamical axiom ($AxN$ or Newton's law of momentum of a mass point), the latter is based upon, and is developed from, two independent dynamical axioms ($Ax1E$ and ...... $2E$ or Euler's laws of momentum and of moment of momentum of rigid body.) On this base in the present paper an analysis of the notion of force in these two main branches of analytical mechanics is proposed. While the forces in mass-point dynamics are represented mathematically by real standard vectors, these vectors are found to be insufficient for the adequate mathematical representation of the forces in ridig body dynamics, for which the arrows (scilicet ordered pairs of normal real standard vectors) prove to be the necessary and sufficcient mathematical tool. These considerations are implements upon the historical background of the crucial moments in the developments of mass-point and rigid body dynamics and of the after-effects of D'Alembert-Langrangean dynamical tradition.