Thiffeault J L, Morrison P J
Institute for Fusion Studies, University of Texas at Austin, 78712-1060, USA.
Ann N Y Acad Sci. 1998 Dec 30;867:109-19. doi: 10.1111/j.1749-6632.1998.tb11253.x.
Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie-Poisson form: the rigid body and the two-dimensional ideal fluid. From these simple cases, we then use the semidirect product extension of algebras to describe more complex physical systems. The Casimir invariants in these systems are examined, and some are shown to be linked to the recovery of information about the configuration of the system. We discuss a case in which the extension is not a semidirect product, namely compressible reduced MHD, and find for this case that the Casimir invariants lend partial information about the configuration of the system.
约化是一个利用对称性来降低哈密顿系统阶数的过程。约化图景中的新变量通常不是正则的：没有明确表示位置和动量的变量，并且得到的泊松括号不是正则类型的。具体来说，我们给出两个产生非正则李 - 泊松形式括号的例子：刚体和二维理想流体。然后，从这些简单情况出发，我们利用代数的半直积扩张来描述更复杂的物理系统。研究了这些系统中的卡西米尔不变量，并且表明其中一些与系统构型信息的恢复有关。我们讨论一种扩张不是半直积的情况，即可压缩约化磁流体动力学，并发现对于这种情况，卡西米尔不变量给出了关于系统构型的部分信息。
