In its conventional formulation, mechanics describes the evolution of states and observables in time. This evolution is governed by a hamiltonian. This is also true for special-relativistic theories, where evolution is governed by a representation of the Poincaré group, which includes a hamiltonian. This conventional formulation is not sufficiently broad because general-relativistic systems – in fact, the world in which we live – do not fit into this conceptual scheme. Therefore we need a more general formulation of mechanics than the conventional one. This formulation must be based on notions of “observable” and “state” that maintain a clear meaning in a general-relativistic context. A formulation of this kind is described in this chapter.

The conventional structure of conventional nonrelativistic mechanics already points rather directly to the relativistic formulation described here. Indeed, many aspects of this formulation are already utilized by many authors. For instance, Arnold identifies the (presymplectic) space with coordinates (t, qi, pi) (time, lagrangian variables and their momenta) as the natural home for mechanics. Souriau has developed a beautiful and little-known relativistic formalism. Probably the first to consider the point of view used here was Lagrange himself, in pointing out that the most convenient definition of “phase space” is the space of the physical motions. Many of the tools used below are also used in hamiltonian treatments of generally covariant theories as constrained systems, although generally within a rather obscure interpretative cloud.