The geometry of Newton's law and rigid systems

Marco Modugno, Raffaele Vitolo

M. Modugno, Department of Applied Mathematics ``G. Sansone", Via S. Marta 3, 50139 Florence, Italy
R. Vitolo, Department of Mathematics ``E. De Giorgi", via per Arnesano, 73100 Lecce, Italy


Abstract. We start by formulating geometrically the Newton's law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. Multi--particle systems are modelled on $n$-th products of the pattern model. We apply the above scheme to discrete rigid systems. We study the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields a new explicit formula for the reaction force in the nodes of the rigid constraint.

AMSclassification. 70G45, 70B10, 70Exx.

Keywords. Classical mechanics, rigid system, Newton's law, Riemannian geometry.