I have seen two formulations of Classical Mechanics:
- Newtonian spacetime (learned it from the lectures of Professor Frederic P. Schuller): Definition: A Newtonian spacetime is a quintuple $(M, \mathcal{O}, \mathcal{A}, \nabla, t)$ where $(M, \mathcal{O}, \mathcal{A}, \nabla)$ is a differentiable manifold with a torsion free connection, and $t\in C^{\infty}(M)$ is such that $dt\neq 0$ and $\nabla dt=0$
- Galilean spacetime (a popular approach used most famously in Arnold's Mathematical Methods of Classical Mechanics) Definition: A Galilean spacetime is a quadruple $(\mathcal{E}, V, g, \tau)$ where $\mathcal{E}$ is an affine space modeled on a four dimesional real vector space $V$, $\tau\in V^*$ and $g$ is an inner product on $\text{ker} \tau$
I find the first approach much more beautiful since it readily extends to relativistic spacetimes. Non the less I have trouble understanding its practical use since I haven't been able to find sources regarding it apart from a series of lectures Klassischen Mechanik by Schuller. On the other hand I've found at least three books that deal with the second approach.
I was wondering two things. Since my German is very bad, does anybody have a source in Spanish or English which deals with the first approach? Moreover, the first approach seems to be more general than the second one. Is there any way to get to the second approach from the first one by introducing some additional structures? I am particularly interested in a claim Schuller made in one of his classes. He said that the idea of having a metric on Newtonian spacetimes is an artifact of inertial reference frames. That's why in the first definition there is no metric, but instead a much weaker object, a connection. Since the second approach has a metric induced by the inner product maybe through a clever choice of charts in the first approach one could arrive to the second one.
Thanks!