Falling coupled oscillators and trigonometric sums

Abstract

A method for evaluating finite trigonometric summations is applied to a system of N coupled oscillators under acceleration. Initial motion of the nth particle is shown to be of the order  2 n 2 T for small time T and the end particle in the continuum limit is shown to initially remain stationary for the time it takes a wavefront to reach it. The average velocities of particles at the ends of the system are shown to take discrete values in a step-like manner.which is a significantly different representation to that found in [5]. While this representation offers no advantage (the representation in [5] offers a reasonably simple and excellent approximation), the method by which it was obtained appears reasonably robust in its applicability. Whereas representations for trigonometric sums, specifically the basic sine related trigonometric sums, contained within [5,6,7] were arrived at using several different methods, ranging in complexity, they may all be obtained using the very simple method discussed in the appendix.

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

“…, d? Specific instances of (3), as well as similar series which include additive or multiplicative characters, have attracted considerable interest, in part because such series appear in different mathematical and physical settings; see, for example, [Do89], [Do92], [CS12] and [Ho18]. As a result, one can see the potential significance in obtaining closed-form evaluations of such sums.…”

Human and automated approaches for finite trigonometric sums

“…The applications go from number theory (see, e.g., [10], but also papers dealing with Dedekind sums, generalizations and associated reciprocity laws (see, e.g., [57,26,6]...) to physics (see, in particular, [20,21,15,16] and the references therein; also see [36]), from enumerative combinatorics (e.g., number of closed walks on a path as in [18]) to binomial identities (see, e.g. [45,40,41,42,36]), and to topology (see, e.g., [37]), etc.…”