A completion of A. Bressan's work on axiomatic foundations of the Mach Painleve` type for various classical theories of continuous media. Part 1: Completion of Bressan's work based on the notion of gravitational equivalence of affine frames.

A paper by Bressan [same journal Cl. Sci. Fis. Mat. Natur. (8) 19 (1987), no. 1, 1–21; MR1006945 (90g:73013)], where various classical theories on continuous bodies are axiomatized from the Mach-Painleve point of view, is completed here in two alternative ways; in that work, ´ among other things, affine inertial frames are defined within classical kinematics. “Here, in Part I, a thermodynamic theory of continuous bodies, in which electrostatic phenomena are not excluded, is dealt with. The notion of gravitational equivalence among affine inertial frames and the notion of gravitational isotropy of these frames are introduced; it is shown that the isotropic inertial frames, gravitationally equivalent to a fixed frame of this kind, are those linked to this by a (possibly improper) Galilean transformation. The Euclidean physical metric on inertial spaces is consequently determined, without introducing it as a primitive notion; and this is the main completion of Bressan’s paper [op. cit.] which is obtained here.