Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
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DOI：
10.1017/s0022377820000331
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发表时间：
2020-02
影响因子：
2.5
通讯作者：
P. Morrison;T. Andreussi;F. Pegoraro
中科院分区：
文献类型：
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作者：
P. Morrison;T. Andreussi;F. Pegoraro
流体流动的不可压缩性约束是拉格朗日在所谓的拉格朗日变量描述中使用他在拉格朗日（变分）公式中的乘子方法施加的。另一种选择是通过使用非正则泊松括号推广狄拉克约束方法，在欧拉变量描述中施加不可压缩性。这里展示了如何使用狄拉克的方法在拉格朗日变量描述中的正则泊松括号和欧拉描述中的非正则泊松括号中施加不可压缩性约束，从而允许密度平流。两种情况下都给出了保体微分同态群上的无限维测地线流动力学，并给出了该动力学的约束条件和原始变量的显式表达式。由于拉格朗日守恒定律和欧拉守恒定律不相同，所以对各种方法进行了比较。
The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.