Cosserat方程的参数化
Parametrization of Cosserat Equations
摘要：
事实上，在工程或数学物理中，许多常微分方程或偏微分方程的解空间“可以/不能”由一定数量的表现为势的任意函数参数化。例如，当且仅当控制系统是可控的并且可以是高阶的时，对于控制系统存在这样的参数化。第一组麦克斯韦方程允许由4-势的一阶参数化。然而，真空中的爱因斯坦方程不允许任何参数化。最后，连续介质力学中的应力方程允许通过n^2（n^2-1）/12个任意函数进行二阶参数化，其中n=2的情况是艾里函数。本文的目的是利用同调代数和代数分析中令人意想不到的深刻结果，首次证明应力/偶应力Cosserat方程允许n^2（n^2-1）/4个任意函数的一阶参数化
英文摘要：
As a matter of fact, the solution space of many systems of ordinary or partial differential equations in engineering or mathematical physics "can/cannot" be parametrized by a certain number of arbitrary functions behaving like potentials. For example, such a parametrization exists for a control system if and only if it is controllable and may be of high order. The first set of Maxwell equations admits a first order parametrization by the 4-potential. However, Einstein equations in vacuum do not admit any parametrization. Finally, the stress equations in continuum mechanics admit a second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the case n=2 being the Airy function. The purpose of this paper is to use unexpected deep results of homological algebra and algebraic analysis in order to prove for the first time that the stress/couple-stress Cosserat equations admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions
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