A Generalized Montgomery Phase Formula for Rotating Self Deforming Bodies

TL;DR

This paper extends Montgomery's phase formula to self deforming bodies with angular momentum, providing a generalized reconstruction method and analytical insights into their motion.

Contribution

It introduces a generalized phase formula for deforming bodies, extending Montgomery's rigid body results, with applications to specific deforming body examples.

Findings

Derived a non-autonomous equation for the shape curve on a sphere.

Provided a reconstruction formula for the deformation phase angle.

Applied the theory to analyze specific deforming body motions.

Abstract

We study the motion of self deforming bodies with non zero angular momentum when the changing shape is known as a function of time. The conserved angular momentum with respect to the center of mass, when seen from a rotating frame, describes a curve on a sphere as it happens for the rigid body motion, though obeying a more complicated non-autonomous equation. We observe that if, after time $\Delta T$, this curve is simple and closed, the deforming body \'{}s orientation in space is fully characterized by an angle or phase $\theta_{M}$. We also give a reconstruction formula for this angle which generalizes R. Montgomery\'{}s well known formula for the rigid body phase. Finally, we apply these techniques to obtain analytical results on the motion of deforming bodies in some concrete examples.