  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.
