Proto-Galilean dynamics of a particle and a continuous body (Q6612635)

The aim of this paper is to formulate the dynamics of a particle and the dynamics of a continuous body in a Proto-Galilean space-time. So, the authors first define a notion of Proto-Galilean space-time which is just a fiber bundle over the time axis; indeed, it is a bundle \(\vartheta: \mathcal{E} \longrightarrow \mathcal{T}\) with typical fiber \(S\) and structure group \(G\). Here, \(\mathcal{T}\) is a one- dimensional differentiable manifold diffeomorphic to the real line \(\mathbb{R}\). Then, every local trivialization of E can be viewed as a local observer or a frame, and the motion of a particle is a section of the fiber bundle. Using this theoretical context, the authors review extensively the basic constructions corresponding to kinematics, force, and stress theory for continuum mechanics on fiber bundles; then, consider the dynamics of a single particle, and finally, present the corresponding dynamics of continuous bodies. In conclusion, this is an interesting paper that provides a theoretical basis for a systematic study of mechanics.