标题： 相对论性连续介质的二阶梯度理论 显示英文标题

标题： Relativistic second gradient theory of continuous media

摘要： 变分相对论是由Souriau在六十年代开发的一种框架，旨在更好地表述广义相对论及其经典极限：经典连续介质力学。它被用来表述广义相对论中的超弹性。在这种情况下，涉及两个主要变量，宇宙（洛伦兹）度量$g$和物质场$\Psi$。然后引入一个依赖于这些变量的1阶喷射的拉格朗日密度，该密度必须满足广义协变性原理。Souriau在1958年证明，在这些假设下，拉格朗日密度仅依赖于物质场$\Psi$的点值和一个次级变量$\mathbf{K}$，即构型，这是一个微分同胚群的不变量，是右柯西-格林张量逆的相对论类比。在本工作中，提出了Souriau结果在广义相对论中二阶梯度理论的扩展。因此，发现了新的高阶微分同胚不变量。它们的经典极限被计算出来，表明三维连续介质力学的二阶梯度理论可以从这样的相对论理论中推导出来。其中一些不变量在伽利略极限中收敛到客观量，另一些则收敛到非客观量。因此，本工作有助于澄清高梯度连续介质力学理论的理论基础。 显示英文摘要

摘要： Variational Relativity is a framework developed by Souriau in the sixties to better formulate General Relativity and its classical limit\,: Classical Continuum Mechanics. It has been used, for instance, to formulate Hyperelasticity in General Relativity. In that case, two primary variables are involved, the universe (Lorentzian) metric $g$ and the matter field $\Psi$. A Lagrangian density depending on the 1-jet of these variables is then introduced which must satisfy the principle of General Covariance. Souriau proved in 1958 that under these hypotheses, the Lagrangian density depends only on the punctual value of the matter field $\Psi$ and of a secondary variable $\mathbf{K}$, the conformation, an invariant of the diffeomorphism group, which is the Relativistic analog of the inverse of the right Cauchy--Green tensor. In the present work, an extension of Souriau's results to a second order gradient theory in General Relativity is presented. Accordingly, new higher order diffeomorphisms invariants are found. Their classical limits are calculated, showing that the 3-dimensional Continuum Mechanics second gradient theory can be derived from such a relativistic theory. Some of these invariants converge to objective quantities in the Galilean limit, others to non-objective quantities. The present work contributes thus to clarify the theoretical foundation of higher gradient Continuum Mechanics theory.