非线性弹性$N$-转子的几何相位：基于Cartan活动标架
Geometric Phases of Nonlinear Elastic $N$-Rotors via Cartan's Moving Frames
AI总结：
本文通过Cartan活动标架研究非线性弹性N-转子的几何相位，揭示形状流形的黎曼结构（双曲平面或Robertson-Walker时空），并推广至自由及受力矩情形，以曲率度量形状接近度。
摘要：
我们研究了具有连续旋转对称性的非线性弹性$N$-转子的几何相位。在哈密顿框架下，相空间的几何结构是一个主纤维丛，即一个底流形（或形状流形）$\mathcal{B}$，以及沿对称方向附着于其上的纤维$\mathcal{F}$。哈密顿动力学的辛结构决定了形状流形的联络和曲率形式。利用零挠率的Cartan结构方程，我们为形状流形找到了一个内蕴的（伪）黎曼度量。我们可以自由地将弹性转子总角动量的旋转符号定义为正或负，例如分别对应逆时针或顺时针，反之亦然。这赋予了底流形$\mathcal{B}$两个不同的度量，两者都与几何相位相容。特别地，当$\mathsf{A}<0$时，度量为伪黎曼度量，且形状流形是具有正曲率的二维Robertson-Walker时空。当$\mathsf{A}>0$时，形状流形是具有负曲率的双曲平面$\mathbb{H}^2$。然后我们将结果推广到自由弹性$N$-转子。我们证明了相关的形状流形$\mathcal{B}$可约化为$(N-1)$个双曲平面$\mathbb{H}^2$（当$\mathsf{A}>0$时）或二维Robertson-Walker时空（当$\mathsf{A}<0$时）的乘积流形，这取决于用于定义总角动量旋转符号的约定。接着我们考虑受时间依赖的自平衡力矩作用的弹性$N$-转子。扩展自治系统的$N$维形状流形具有与自由弹性转子的$(N-1)$维形状流形类似的结构。形状流形的黎曼结构提供了在曲率或诱导几何相位方面一个形状与另一个形状接近程度的内蕴度量。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
