Discrete Harmonic Oscillator: Evolution of Notation and Cumulative Erratum

Abstract

As the project on the discrete harmonic oscillator advanced, the notation evolved in parallel with our understanding of the problem, and it became progressively clear that a change of notation concerning the natural angular frequency of oscillation of the system, and related parameters, was required in order to conform to the spirit of the physics being developed. Specifically the notation needed to reflect the fact that the discrete harmonic oscillator is the more general entity, with the continuous harmonic oscillator being a special (and to some extent unrealistic) limiting case corresponding to Δt→0, where Δt is the discretizing time interval. Furthermore, as the discrete nature of time (and eventually space), incursion (anticipation), system bifurcation, hyperincursion, frequency dependent correlation, and synchronization, where progressively integrated into the fabric of the discretizing algorithm, a new discrete, perfectly stable and energy conserving, formalism emerged, and with it evolved, in parallel, the expression for the total energy. In this short note we will focus on the evolution, throughout the project, of the notation for frequency, and the expression for the discrete total energy. We also provide a cumulative erratum for the seven papers in the project so far, as well as a table summarizing the evolution of the notation for the angular frequency and related parameters.