Geodesic
A completion of A. Bressan's work on axiomatic foundations of the Mach Painlevé type for various classical theories of continuous media. Part 1. Completion of Bressan's work based on the notion of gravitational equivalence of affine inertial frames
Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 1, pp. 35-46.

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The work [3], where various classical theories on continuous bodies are axiomatized from the Mach-Painlevè 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 [3] which is obtained here.
Si completa in due maniere alternative il lavoro [3] di assiomatizzazione alla Mach Painlevé di varie teorie classiche di sistemi continui, ove, tra l'altro, riguardo alla cinematica classica si arriva a definire i riferimenti inerziali affini. Nella Parte 1 si considera una generica teoria termodinamica dei sistemi continui nella quale non si escludono fenomeni elettrostatici. Si introducono le nozioni di equivalenza gravitazionale tra riferimenti inerziali affini e di isotropia gravitazionale di tali riferimenti; si dimostra che tutti e soli i riferimenti inerziali isotropi gravitazionalmente equivalenti ad un fissato tale riferimento sono legati a questo da una trasformazione Galileiana (impropria). La metrica fisica Euclidea sugli spazi inerziali risulta quindi determinata, senza bisogno di introdurla come nozione primitiva; e ciò costituisce il principale completamento di [3] qui conseguito. 
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Montanaro, Adriano. A completion of A. Bressan's work on axiomatic foundations of the Mach Painlevé type for various classical theories of continuous media. Part 1. Completion of Bressan's work based on the notion of gravitational equivalence of affine inertial frames. Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 1, pp. 35-46. http://geodesic.mathdoc.fr/item/RLINA_1987_8_81_1_a5/

[1] Bressan Aldo (1962) - Metodo di assiomatizzazione in senso stretto della Meccanica classica, «Rend. Sem. Mat. Univ. Padova», 32, 55-212. | fulltext EuDML | fulltext mini-dml | Zbl

[2] Bressan Aldo e Montanaro Adriano (1982) - Axiomatic foundations of the kinematics common to classical physics and special relativity, «Rend. Sem. Mat. Univ. Padova», 68, 7-26. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[3] Bressan Aldo - Towards axiomatic foundations of the Mach Painlevé type for classical or relativistic theories of continuous media. Printed in the present review. | Zbl

[4] Kellogg Oliver Dimon (1929) - Foundations of potential theory. Berlin, Verlag Von Julius Springer. | fulltext EuDML | MR

[5] Montanaro Adriano - A completion of A. Bressan's work on axiomatic foundations of the Mach Painlevé type for various classical theories of continuous media. Part 2: Alternative completion of Bressan's work, fit for extension to special relativity. Being printed. | Zbl