A rigorous proof for the equivalence of the projective Newton–Euler equations and the Lagrange equations of second kind for spatial rigid multibody systems

Abstract

It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations of motion for holonomic rigid multibody systems. It can be shown that both approaches follow from the principles of d’Alembert or Jourdain. However, as to the author’s knowledge, no direct rigorous proof for the equivalence of these approaches is given in literature so far when it comes to spatial systems of rigid bodies. In this paper, we present a novel proof that directly addresses the projective Newton–Euler equation and the Lagrange equation of second kind without the detour via variational principles. The proof is mainly based on vector and matrix manipulations and elementary concepts of differential geometry. Although the mathematical framework is thereby kept simple, the argumentation is considerably more complex compared to the case of planar systems of rigid bodies or spatial systems of particles. An illustrative example is presented.