Free fall motion in an invariant field of forces: the 2D-case.

Following [12] and [14], the Lie groups may be classified into four families, with respect to the property of the invariant vector fields to be geodesic. A detailed study in dimensions 2 and 3 ([13]) led to many examples of particular invariant vector fields ξ geometrized, in this manner, by means of some properly chosen invariant Riemannian metrics g. In this paper we extend the previous investigation, and analyse the behavior of all the invariant vector fields ξ and of all their associated invariant Riemannian metrics g. The 2-dimensional case is completely solved: given a left invariant field of forces ξ on a 2-dimensional Lie group G, we determine all the metrics g 2 Riem(G; ξ) such that ξ becomes geodesic in (G; g) and all the possible trajectories of free fall particles moving in the "Universe" (G; g). [ABSTRACT FROM AUTHOR]