Point Motion in Flat Spaces: An Ample Starting Point

In this work, we approach the interesting problem of representing and studying the position, velocity, acceleration and arc-length of trajectories of points defined in a two-dimensional geometrical space, which has been chosen for being simpler than three-dimensional spaces while retaining much of their structure, and richer than the one-dimensional line. In addition to providing an integrated revision of several aspects from linear algebra, differential calculus, and analytic geometry, the present work can also be understood as a starting point for studies of several ares in physics, especially particle dynamics, but also including electromagnetism, fluid dynamics and statistical physics. The presentation starts with mathematical concepts of spaces, bases and vectors, as well as the Dirac delta 'function' and its derivative, as a means for approaching motion discontinuity. The text then proceeds by presenting the concepts of position, motion (or trajectory) of points in terms of parametric curves, also addressing. The concept of arc-length of a trajectory is then presented, followed by a discussion of how point motion can interact with scalar and vector fields (including the concepts of gradient and line integrals). Brief presentations of the the concepts of relative position and speed, as well as moving frames conclude this work. Several examples and illustrations of the covered concepts have also been provided