Differential constants of motion for systems of free gravitating particles. I. Newton's theory

Differential constants of motion for systems of free gravitating particles in the Newtonian frame are first defined and then determined. It is shown that they are all implied by the existence of the first integral invariants of Poincaré known from classical mechanics, or by the circulation theorem known from hydrodynamics. It is proved further that the restriction to vacuum conditions does not change the set of differential constants of motion. Another consequence is that nothing can be inferred from local (in space and time) measurements about the displacement, velocity, and orientation of a laboratory in free fall relative to a fixed Galilean frame.