Minimum Parametrization of the Cauchy Stress Operator

• 作者: J.-F. Pommaret
• 来源期刊: 现代物理(英文)
• 学科: 数学
• 关键词: Differential Operator Differential Sequence Killing Operator Riemann Operator Bianchi Operator Cauchy Operator Electromagnetism Elasticity General Relativity Gravitational Waves
• 年，卷（期）: 2021,（4）
• 页码范围: 453-482
• 页数: 30页
• 分类号: O17

摘要

When D: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&xi;</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&rarr;</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&eta;</span></span></em><em><span style="white-space:nowrap;"></span></em><em></em></span> </span>is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D₁: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&eta;</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&rarr;</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&xi;</span> </span></em></span></span>such that <span style="white-space:nowrap;">D<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&xi;</span></span></em></span>=<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&eta;</span></span></em></span></span> implies <span style="white-space:nowrap;">D₁<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">&eta;</span></span></em></span>=0</span>. When D is involutive, the procedure provides successive first order involutive operators D1, ..., D_n, when the ground manifold has dimension <em>n</em>, a result first found by M. Janet as early as in 1920, in a footnote. However, the link between this “Janet sequence” and the “Spencer sequence” first found by the author of this paper in 1978 is still not acknowledged. Conversely, when D₁ is given, a more difficult “inverse problem” is to look for an operator D: <span style="white-space:nowrap;"><span style="white-space:nowrap;">