Symplectic geometry and the inertial principle

The possibility of introducing a symplectic inertial principle which is compatible with Special Relativity is explored. Inertial states are represented as leaves of a real polarization X. An almost pseudo-Kaehler structure compatible with the symplectic structure and the decomposition of M by X and Y is constructed in M. An inertial principle is introduced as a restriction on the curvature which prescribes how observers in distinct frames correlate their observations of events. Next it is shown that a model structure consistent with Special Relativity is a solution of these equations. This model can be constructed at least locally in the cotangent bundle T/sup */N of a spacetime N, and gives an extension of Special Relativistic effects to nonaffine geometries. The physical effects of nonvanishing curvature are considered. This construction allows the introduction of a Doppler effect into Robinson-Walker Universes. It also implies the existence of a generalized Poincare group where in the semi-direct product structure of boost and translations is modified by curvature. Finally, following Souriau (7) a definition of force is introduced and it is shown that the model structure naturally yields Newton's equations. This fact is taken as evidence that the restriction on the curvature does in fact function as an inertial principle