The geometry of Newton's law and rigid systems

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. In fact, we use this scheme for further generalisation devoted to a constrained particle, to a discrete system of several free and constrained particles. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. In the second case, we obtain an explicit formula for the reaction force via the second fundamental form of the constrained configuration space. For multi--particle systems we describe geometrically the splitting related to the center of mass and relative velocities; in this way we emphasise the geometric source of classical formulas. Then, the above scheme is applied in detail to discrete rigid systems. We start by analysing the geometry of the rigid configuration space. In this way we recover the classical formula for the velocity of the rigid system via the parallelisation of Lie groups. Moreover, we study in detail 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 an explicit formula for the reaction force in the nodes of the rigid constraint.