Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow

Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow
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柯西几乎被遗忘的 3D 不可压缩流欧拉方程的拉格朗日公式

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发表时间：
2014
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影响因子：
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通讯作者：
B. Villone
B. Villone
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作者：
U. Frisch;B. Villone

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摘要 两篇珍贵的论文，一篇是奥古斯丁·柯西于1815年提交给法国科学院的， 另一个是赫尔曼·汉克尔于1861年提交给哥廷根大学，包含主要内容 涡度动力学的发现，其影响正在迅速增加。柯西发现了一个 用三个不变量表示的三维理想不可压缩流的拉格朗日公式， 把现在熟知的涡量守恒定律沿着推广到三维空间 二维流动的流体粒子轨迹。这是最近才被用来 三维不可压欧拉流中流体质点轨迹时间解析性证明 并且可以扩展到可压缩流，特别是宇宙学暗物质。 汉克尔表明，柯西的公式给出了一个非常简单的拉格朗日推导， Helmholtz涡量通量不变量，并在证明的中间，推导出一个 中间结果，这是周围的速度循环守恒 封闭轮廓随流体移动。这个循环定理将被重新发现 由威廉·汤姆森（Kelvin）于1869年独立完成。柯西不变量只是 偶尔在19世纪被引用-除了汉克尔，最重要的是乔治斯托克斯和 莫里斯·莱维--在20世纪更是如此，直到他们通过艾美奖被重新发现。 Noether定理在1960年末被提出，但直到20世纪末才被重新归因于Cauchy 世纪的俄罗斯科学家。
Abstract Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to Göttingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy’s formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy’s invariants were only occasionally cited in the 19th century – besides Hankel, foremost by George Stokes and Maurice Lévy – and even less so in the 20th until they were rediscovered via Emmy Noether’s theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.