Geometric Phases of Nonlinear Elastic $N$-Rotors via Cartan's Moving Frames

TL;DR

This paper investigates the geometric phases of nonlinear elastic N-rotors using Cartan's moving frames, revealing the shape manifold's structure as a pseudo-Riemannian or hyperbolic space, with implications for understanding their intrinsic geometric properties.

Contribution

It introduces a geometric framework for elastic N-rotors, deriving the shape manifold's metric and curvature, and generalizes to systems with time-dependent moments, connecting geometry with physical dynamics.

Findings

Shape manifold is a 2D Robertson-Walker spacetime or hyperbolic plane.

The shape space's metric depends on the rotation sign convention.

The Riemannian structure measures shape similarity via curvature and geometric phase.

Abstract

We study the geometric phases of nonlinear elastic $N$-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold~$\mathcal{B}$, and fibers $\mathcal{F}$ along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan's structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. One has the freedom to define the rotation sign of the total angular momentum of the elastic rotors as either positive or negative, e.g., counterclockwise or clockwise, respectively, or viceversa. This endows the base manifold~$\mathcal{B}$ with two distinct metrics both compatible with the geometric phase. In particular, the metric is pseudo-Riemannian if $\mathsf{A}<0$, and the shape manifold is a $2$D~Robertson-Walker spacetime with positive curvature. For $\mathsf{A}>0$, the shape manifold is the hyperbolic plane $\mathbb{H}^2$ with negative curvature. We then generalize our results to free elastic $N$-rotors. We show that the associated shape manifold~$\mathcal{B}$ is reducible to the product manifold of $(N-1)$ hyperbolic planes $\mathbb{H}^2$~($\mathsf{A}>0$), or $2$D~Robertson-Walker spacetimes~($\mathsf{A}<0$) depending on the convection used to define the rotation sign of the total angular momentum. We then consider elastic $N$-rotors subject to time-dependent self-equilibrated moments. The $N$-dimensional shape manifold of the extended autonomous system has a structure similar to that of the $(N-1)$-dimensional shape manifold of free elastic rotors. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase.