Smooth Evolution: An Examination of Dynamics in Smooth Worlds

Throughout the history of physics a key tool in the development of physical laws has been the infinitesimal. Infinitesimals entered the scene with the development of the calculus by Newton and Leibniz and went on to be used, to mention some of the most notable cases, in Euler’s derivation of the laws of hydrodynamics, in the development of analytical mechanics by Euler, Lagrange and Hamilton and in Riemann’s differential geometry. Infinitesimals are a key conceptual tool for constructing laws in which local behaviour determines global behaviour. This is a natural way of deriving physical laws, which explains why infinitesimal arguments are so popular in the history of physics. A limitation of these methods, however, was that they were not rigorous, a fact that was more of a concern to 19th Century mathematicians, rather than the physicists. The attempts to rigorize the calculus, which culminated in the development of set theory, replaced infinitesimals with limits and rendered infinitesimals mathematical fictions, since, in a classical context, physical models use archimedean fields so the only number that has the properties that infinitesimals have is 0. This did not stop physicists from using infinitesimal arguments in the development of physical laws, however, since such methods much more natural then the use of limits.