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

The unity of instantaneous spectral moments and physical moments