Summary

We study the main structures of General Relativistic one-particle Dynamics, both from an absolute and an observed (by a frame of reference) point of view. This approach is based maninly on the concept of pseudoconnection on the tangent space, which enables us to arrange several types of derivation, and on the essential use of the "position space"; this is the set of all particles of a continuum, which works as a frame, and has a natural manifold structure. In comparison with the usual "projection method", we obtain a more geometrical picture.

Introduction [to be typeset by Plain TeX]

Relativistic Dynamics can be formulated in an absolute (i.e. independent of frame) way, or as observed by a frame of reference; the second way is more intricate but is important for physical interpretation, therefore several authors have worked on it; their method is essentially based on the decomposition of tensors with respect to the velocity of the frame.

Our approach is distinguished by the essential use of the "position space" $P$ (i.e. the set of all world-lines of the frame, which has a natural manifold structure) and of spaces derived from $P$. On these spaces we can define some structures which give rise to equivalent formulations of the law of motion. The most important of these structures are "pseudoconnections"; pseudoconnections are more general structures than connections since some linearity properties are dropped; they will probably grow in importance for field theories. A number of pseudoconnections is defined in a natural way on $P$, each of them giving rise to a derivation (some of these derivations correspond substantially to known ones, like the Fermi derivative). Then a clear picture of observed Kynematics and Dynamics arises from an original reformulation of the traditional projection method, and each structure is given a clear physical interpretation. One realizes that the concept of frame is very different from that of coordinate system, and that often the physical and geometrical meaning of coordinate formulas is not sufficiently clear, mainly with regard to derivations.

We think that the most simple and direct way to handle connections and pseudoconnections is to see them as structures on the tangent space, rather than as covariant derivatives or structures on a principal bundle. More precisely, a pseudoconnection is a splitting of the tangent space of a bundle into a vertical and a horizontal part, many objects of observed Dynamics are unified by this concept and are given a geometrical interpretation.

Moreover we give equivalent formulations of Dynamics in terms of other structures on $P$ such as contact structures and sprays; however we think that the pseudoconnection way is the most simple and clear one, both from a conceptual and practical point of view.