用变分法求解移动沙发问题
Solving Moving Sofa Problem Using Calculus of Variations
AI总结：
本文将移动沙发问题表述为变分问题，通过Euler-Lagrange方程刻画最优形状，并数值复现Gerver常数，同时验证Gerver沙发与Romik汽车满足最大面积必要条件。
摘要：
1966年，Leo Moser提出了“移动沙发问题”，旨在确定能够通过单位宽度90度走廊进行移动的形状的最大面积。该问题至今仍未解决并保持开放。本文采用变分法来求解该问题。在假设轨迹和包络线均为凸的条件下，沙发面积被表述为一组曲线参数方程上的积分泛函。最终形状通过求解Euler-Lagrange方程确定。利用数值方法，我们得到非平凡面积2.2195316，这与自1992年以来广为人知的Gerver常数一致。我们证明，Gerver沙发和Romik汽车的结果都满足最大面积必要条件的Euler-Lagrange方程。我们还探索了其他情形和非对称条件，并讨论了其他变体问题。
英文摘要：
In 1966, Leo Moser introduced the "moving sofa problem," which seeks to determine the largest area of a shape that can be maneuvered through a 90-degree hallway of unit-width. This problem remains unsolved and open yet. In this paper, we employ calculus of variations method to solve this problem. Assuming the trajectories and envelopes are convex, the sofa's area is formulated as an integral functional on a set of parametric equations for curves. The final shape is determined by solving the Euler-Lagrange equations. Utilizing numerical methods, we obtain the non-trivial area of 2.2195316, consistent with the previously well-known Gerver's constant since 1992. We prove that both the results of Gerver's sofa and Romik's car satisfy the Euler-Lagrange equations for the necessary condition of maximal area. We also explore additional cases and asymmetric conditions, and discuss other variant problems.
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