From Elastica to Floating Bodies of Equilibrium
From Elastica to Floating Bodies of Equilibrium
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从弹性体到浮动平衡体
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
F. Wegner
中科院分区：
文献类型：
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作者：
F. Wegner
对二维平衡浮体和自行车问题有关的曲线作了简短的历史叙述。Bor, Levi， Perline和Tabachnikov发现，有相当一部分已经被伯努利和欧拉描述为弹性力学，也被Levy和Halphen描述为压力下的弹性力学。奥尔巴赫已经意识到辛德勒已经为浮体问题描述了曲线。一个更大的曲线类解决了自行车问题。随后的章节处理了一些补充细节：给出了弹性材料和压力下弹性材料方程的几个推导。考虑了Zindler曲线的性质和其他数学家在平衡浮体问题上的一些工作。压力作用下弹性的特殊情况可以简化为代数曲线，如Greenhill所示。由于这里考虑的大多数曲线都是自行车曲线，所以在此添加一些关于它们的注释。
A short historical account of the curves related to the two-dimensional floating bodies of equilibrium and the bicycle problem is given. Bor, Levi, Perline and Tabachnikov found, quite a number had already been described as Elastica by Bernoulli and Euler and as Elastica under Pressure or Buckled Rings by Levy and Halphen. Auerbach already realized that Zindler had described curves for the floating bodies problem. An even larger class of curves solves the bicycle problem. The subsequent sections deal with some supplemental details: Several derivations of the equations for the elastica and elastica under pressure are given. Properties of Zindler curves and some work on the problem of floating bodies of equilibrium by other mathematicians are considered. Special cases of elastica under pressure reduce to algebraic curves, as shown by Greenhill. Since most of the curves considered here are bicycle curves, a few remarks concerning them are added.