Gauge theory of the falling cat
Gauge theory of the falling cat
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落猫的规范理论
DOI：
10.1090/fic/001/09
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
R. Montgomery
中科院分区：
文献类型：
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作者：
R. Montgomery
Kane 和 Scher [18] 提出了一个机械模型，以解释和更好地理解坠落的猫如何自我平衡。他们的模型猫由两个相同的轴对称刚体组成，这两个刚体通过特殊的“无扭转”关节连接在一起。第一个问题是模型猫在自由落体时如何将自己摆正，从倒置位置开始没有角动量。 Kane 和 Scher，以及更早的 Rademaker 和 ter Braak [25]' 为此提出了一个具体策略。但他们没有研究寻找执行翻转的一般策略的问题。第二个问题是以最佳方式表演她的技巧。这些可以被视为控制理论中的问题。在早期的论文[22]、[21]中，我们发展了一种关于零角动量自由落体中可变形物体的姿态或方向控制的一般理论。这些论文是 Wilczek 和 Shapere [27] 以及 Guichardet [12] 工作的成果。这些早期工作的要点是，可以在物理学家和数学家的规范理论与可变形体的方向控制中出现的问题之间建立一个字典。简而言之，在这本词典中，身体形状的空间扮演着基础空间的角色，或者说是物理学家规范理论中的时空。它的切线空间就是控制空间。状态空间或物体的配置空间是该理论的主束。规范组是身体的刚性重新定向组。规范场概括了角动量为零的条件。本文的目的是将我们的一般理论应用于 KaneScher 猫。如果没有特殊的无扭转关节，形状空间是群 80(3)，其中的一个元素表示猫的一半相对于固定到另一半的框架的姿态。配置空间为 Q = 80(3) x 80(3)，每个半体有一个 80(3)，规格组为 80(3)，对角地作用于配置空间。这是我们的主要结果。
Kane and Scher [18] proposed a mechanical model in order to explain and better understand how a falling cat rights herself. Their model cat consists of two identical axi-symmetric rigid bodies which are joined by a special 'no-twist' joint. The first problem is for the model cat to right herself while in freefall with no angular momentum beginning from an upside-down position. Kane and Scher, and earlier Rademaker and ter Braak [25]' proposed a specific strategy for doing this. But they did not study the problem of finding the general strategy for performing the flip. A second problem is for the to perform her trick in an optimal way. These can be viewed as problems in control theory. In earlier papers [22], [21] we developed a general theory for the attitude, or orientation control, of deformable bodies in freefall with zero angular momentum. These papers were outgrowths of work by Wilczek and Shapere [27] and Guichardet [12] . The main point of these earlier works is that a dictionary can be developed between the gauge theory of the physicist's and mathematicians, and the problems occuring in the orientation control of deformable bodies. Briefly, in this dictionary the space of shapes of the body plays the role of the base space, or space-time in the physicist's gauge theory. Its tangent space is the space of controls. The state space, or configuration space of the body, is principal bundle of the theory. The gauge group is the group of rigid reorientations of the body. The gauge field summarizes the condition that the angular momentum be zero. The purpose of the present paper is to apply our general theory to the KaneScher cat. Without the special no-twist joint, the shape space is the group 80(3) with an element in it representing the attitude of one half of the cat relative to a frame fixed to the other. The configuration space is Q = 80(3) x 80(3) with one 80(3) for each body half, and the gauge group is 80(3), acting diagonally on the configuration space. Here are our main results.