零角动量下的等边五角大楼：通过最佳变形实现最大旋转
The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation Through Optimal Deformation
摘要：
平面上边长固定的五边形具有二维形状空间。把五边形看作是一个角上有点质量的力学系统，我们回答了五边形在零角动量的情况下能旋转多少的问题。我们证明了等边五边形的形空间具有亏格4，并通过关于对称群D_5的离散对称约化找到了一个基本区域。在零角动量时，形状空间中的环的旋转量Δθ被解释为几何相位，并且被获得为函数B在由环包围的形状空间区域上的积分。用一个简单的变分变元，我们确定局部最优环作为函数B的零轮廓。所得到的形状变化被表示为傅里叶级数，并且对于围绕规则五角星形的环找到Δθ \approx 45 \degree的全局极大值。我们还表明，限制允许的形状凸五边形的最佳循环是凸区域的边界，并给出Δθ \approx 19 \degree。
英文摘要：
A pentagon in the plane with fixed side-lengths has a two-dimensional shape space. Considering the pentagon as a mechanical system with point masses at the corners we answer the question of how much the pentagon can rotate with zero angular momentum. We show that the shape space of the equilateral pentagon has genus 4 and find a fundamental region by discrete symmetry reduction with respect to symmetry group D_5. The amount of rotation Δθfor a loop in shape space at zero angular momentum is interpreted as a geometric phase and is obtained as an integral of a function B over the region of shape space enclosed by the loop. With a simple variational argument we determine locally optimal loops as the zero contours of the function B. The resulting shape change is represented as a Fourier series, and the global maximum of Δθ\approx 45\degree is found for a loop around the regular pentagram. We also show that restricting allowed shapes to convex pentagons the optimal loop is the boundary of the convex region and gives Δθ\approx 19\degree.
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