تقرير
Cauchy, Cosserat, Clausius, Maxwell, Weyl Equations Revisited
| العنوان: | Cauchy, Cosserat, Clausius, Maxwell, Weyl Equations Revisited |
|---|---|
| المؤلفون: | Pommaret, J. -F. |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematical Physics, Mathematics - Group Theory, 14L30, 16E05, 22E70, 53A30, 53B21, 58H05, 78A25 |
| الوصف: | The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870), the Maxwell/Weyl equations (1873,1918) are among the most famous partial differential equations that can be found today in any textbook dealing {\it separately and/or successively} with elasticity theory, continuum mechanics, thermodynamics, electromagnetism and electrodynamics. Over a manifold of dimension $n$, their respective numbers are $n, n(n-1)/2, 1, n$ with a total of $(n+1)(n+2)/2$, that is $15$ when $n= 4$ for space-time. As a matter of fact, this is just the number of parameters of the Lie group of conformal transformations with $n$ translations, $n(n-1)/2$ rotations, $1$ dilatation and $n$ highly non-linear elations introduced by E. Cartan in $1922$. The purpose of this short but difficult paper is to prove that the form of these equations only depends on the structure of the conformal group for $n\geq 1$ arbitrary because they are described {\it as a whole} by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained for the first time by totally new differential geometric methods. Meanwhile, these equations can be all parametrized by the adjoint of the second Spencer operator through $ n(n^2 - 1)(n+2)/4$ potentials.This result brings the need to revisit the mathematical foundations of Electromagnetism and Gauge Theory according to a clever but rarely quoted paper of H. Poincar\'{e} (1901). Comment: In classical Gauge Theory, the group U(1) is not acting on space-time when describing Maxwell equations. On the contrary, in this new approach, a Lie group of transformations is considered as a Lie pseudogroup of transformations used in order to construct differential sequences and their adjoint sequences. arXiv admin note: text overlap with arXiv:2007.01710, arXiv:1504.04118 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2401.14563 |
| رقم الأكسشن: | edsarx.2401.14563 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |