1201.5916 (Jonathan M. Lilly)

The unity of instantaneous spectral moments and physical moments    [PDF]

Jonathan M. Lilly

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.

View original: http://arxiv.org/abs/1201.5916