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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Vorticity and symplecticity in multisymplectic, Lagrangian gas dynamics
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Vorticity and symplecticity in multisymplectic, Lagrangian gas dynamics

机译：多辛拉格朗日气体动力学中的涡性和辛性

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摘要

The Lagrangian, multi-dimensional, ideal, compressible gas dynamic equations are written in a multi-symplectic form, in which the Lagrangian fluid labels, m(i) (the Lagrangian mass coordinates) and time t are the independent variables, and in which the Eulerian position of the fluid element x = x(m, t) and the entropy S = S(m, t) are the dependent variables. Constraints in the variational principle are incorporated by means of Lagrange multipliers. The constraints are: the entropy advection equation S-t = 0, the Lagrangian map equation x(t) = u where u is the fluid velocity, and the mass continuity equation which has the form J = tau where J = det(x(ij)) is the Jacobian of the Lagrangian map in which x(ij) = partial derivative x(i)/partial derivative m(j) and tau = 1/rho is the specific volume of the gas. The internal energy per unit volume of the gas epsilon = epsilon(rho, S) corresponds to a non-barotropic gas. The Lagrangian is used to define multi-momenta, and to develop de Donder-Weyl Hamiltonian equations. The de Donder-Weyl equations are cast in a multi-symplectic form. The pullback conservation laws and the symplecticity conservation laws are obtained. One class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws, and another class of symplecticity laws are related to derivatives of the Lagrangian energy conservation law with respect to the Lagrangian mass coordinates mi. We show that the vorticity-symplecticity laws can be derived by a Lie dragging method, and also by using Noether's second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action. We obtain the Cartan-Poincare form describing the equations and we discuss a set of differential forms representing the equation system.
机译：拉格朗日的，多维的，理想的，可压缩的气体动力学方程以多符号形式表示，其中拉格朗日流体标记m（i）（拉格朗日质量坐标）和时间t是自变量，其中流体元素的欧拉位置x = x（m，t）和熵S = S（m，t）是因变量。借助Lagrange乘子将变分原理中的约束合并在一起。约束条件是：熵对流方程St = 0，拉格朗日映射方程x（t）= u，其中u是流体速度，质量连续性方程，形式为J = tau，其中J = det（x（ij） ）是拉格朗日图的雅可比行列式，其中x（ij）=偏导数x（i）/偏导数m（j），tau = 1 / rho是气体的比容。气体epsilon = epsilon（rho，S）的每单位体积的内部能量对应于非正压气体。拉格朗日用于定义多动量，并发展de Donder-Weyl Hamiltonian方程。 de Donder-Weyl方程以多符号形式转换。得到了回撤守恒律和辛辛守恒律。一类辛辛守恒律引起涡度和潜在的涡度类型守恒律，而另一类辛辛律则与拉格朗日能量守恒律关于拉格朗日质量坐标mi的导数有关。我们表明，涡度-辛律定律可以通过Lie拖动方法来推导，也可以通过使用Noether第二定理和流体重标记对称性（即作用的发散对称性）来推导。我们获得描述方程的Cartan-Poincare形式，并讨论代表方程系统的一组微分形式。

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