Quasi periodic motions from Hipparchus to Kolmogorov

Contemporary research on the problem of chaotic motions in dynamical systems finds its roots in the Aristotelian idea, often presented as kind of funny in high schools, that motions can always be considered as composed by circular uniform motions,. The reason of this conception is the perfection and simplicity of such motions (of which the uniform rectilinear motion case must be thought as a limit case). The idea is far older than Hipparchus (Nicea, 194-120 b.c.) from whom, for simplicity of exposition it is convenient to start (in fact, the epicycle appears at least with Apollonius (Perga, 240–170 b.c)). The first step is to understand exactly what the Greeks really meant for motion composed by circular uniform motions. This indeed is by no means a vague and qualitative notion, and in Greek science it acquired a very precise and quantitative meaning that was summarized in all its surprising rigor and power in the Almagest of Ptolemy (Alexandria, ∼100-175 a.d.). We thus define the motion composed by n uniform circular motions with angular velocities ω1, . . . , ωn that is, implicitly, in use in the Almagest, but following contemporary mathematical terminology. A motion is said to be quasi periodic if every coordinate of any point of the system, observed as time t varies, can be represented as: