The unity of instantaneous spectral moments and physical moments

Abstract: A modulated oscillation in two or three dimensions can be represented as the trajectory traced out in space by a particle orbiting an ellipse, the properties of which vary as a function of time. Generalizing ideas from signal analysis, the signal variability can be described in terms of kinematic quantities, the instantaneous moments, that formalize our intuitive notions of time-varying frequency and amplitude. On the other hand, if we observed an ellipse evolving in space we would seek to describe it in terms of its physical moments, such as angular momentum, moment of inertia, etc. The main result of this paper is to show that the two sets of moments are identical. Most significantly, an essential physical quantity---the circulation---is the same as the product of the two most important kinematic quantities, the instantaneous frequency and the squared instantaneous amplitude. In addition to providing a rich set of geometric tools for the analysis of nonstationary oscillations in two or three dimensions, this result also has implications for the practical problem of inferring physical ellipse parameters from the trajectory of a single particle on the ellipse periphery, as is frequently encountered in the study of vortex motions.