13.3.1 The Newtonian Model

The most basic formulation of mechanics goes back to Newton and Euler, and parts of it are commonly studied in basic physics courses. Consider a world ${\cal W}$ defined as in Section 3.1, except here a 1D world ${\cal W}= {\mathbb{R}}$ is allowed, in addition to 2D and 3D worlds. A notion of time is also needed. The space of motions that can be obtained in the space-time continuum can be formalized as a Galilean group [39]; however, the presentation here will utilize standard intuitive notions of time and Euclidean space. It is also assumed that any relativistic effects due to curvature of the time-space continuum are nonexistent (Newton and Euler did not know about this, and it is insignificant for most small-scale mechanical systems on or near the earth).