多辛拉格朗日，一维气体动力学
Multi-Symplectic Lagrangian, One-Dimensional Gas Dynamics
摘要：
以拉格朗日质量坐标$m$和时间$t$为自变量，流体单元$x=x(m,t)$的欧拉位置为因变量，将理想一维（1D）可压缩气体动力学的拉格朗日方程写成多辛形式.这种方法不同于使用Clebsch变量的欧拉多辛方法。拉格朗日约束用于指定与拉格朗日映射一致的$x_m$、$x_t$和$S_t$的方程，其中$S$是气体的熵。我们要求$S_t=0$对应于熵$S$随流动的平流。我们表明，拉格朗日哈密顿方程的de Donder-Weyl多动量公式。讨论了拉回守恒律和辛性守恒律。拉回守恒定律对应于作用相对于时间平移（能量守恒）和诺特定理中$m$中平移的不变性。由于$m$平移不变性的守恒律产生了一个新的非局部守恒律，涉及Clebsch变量$r$用于施加$\partial S(m,t)/\partial t=0$。在Noether定理中关于$x$的平移不变性与动量守恒有关。我们得到了系统的Cartan-Poincaré形式，并利用它得到了表示方程组的两种形式的封闭理想。
英文摘要：
The equations of Lagrangian, ideal, one-dimensional (1D), compressible gas dynamics are written in a multi-symplectic form using the Lagrangian mass coordinate $m$ and time $t$ as independent variables, and in which the Eulerian position of the fluid element $x=x(m,t)$ is one of the dependent variables. This approach differs from the Eulerian, multi-symplectic approach using Clebsch variables. Lagrangian constraints are used to specify equations for $x_m$, $x_t$ and $S_t$ consistent with the Lagrangian map, where $S$ is the entropy of the gas. We require $S_t=0$ corresponding to advection of the entropy $S$ with the flow. We show that the Lagrangian Hamiltonian equations are related to the de Donder-Weyl multi-momentum formulation. The pullback conservation laws and the symplecticity conservation laws are discussed. The pullback conservation laws correspond to invariance of the action with respect to translations in time (energy conservation) and translations in $m$ in Noether's theorem. The conservation law due to $m$-translation invariance gives rise to a novel nonlocal conservation law involving the Clebsch variable $r$ used to impose $\partial S(m,t)/\partial t=0$. Translation invariance with respect to $x$ in Noether's theorem is associated with momentum conservation. We obtain the Cartan-Poincaré form for the system, and use it to obtain a closed ideal of two-forms representing the equation system.
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