Hamiltonian formalism for nonlinear waves
Hamiltonian formalism for nonlinear waves
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DOI：
10.1070/pu1997v040n11abeh000304
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发表时间：
1997-11
期刊：
影响因子：
2.7
通讯作者：
V E Zakharov;Evgenii A Kuznetsov
中科院分区：
文献类型：
--
作者：
V E Zakharov;Evgenii A Kuznetsov
回顾了流体动力学类型系统在等离子体、流体动力学和磁流体动力学中的应用的哈密顿描述，重点是引入规范变量的问题。指出了与其他哈密顿方法的关系，特别是自然变量泊松括号。结果表明，非正则泊松括号的简并性与一种特殊类型的对称性有关，即流体粒子拉格朗日标记的重新标记变换，所有已知的涡度守恒定理，如 Ertel 定理、Cauchy 定理、Kelvin 定理，以及涡度冻结定理和拓扑 Hopf 不变量都源自该定理。描述了典型变量在无碰撞等离子体动力学中的应用。讨论了 Benney 方程和 Rossby 波动方程的哈密顿结构。 Davey-Stewartson 方程采用哈密顿形式。基于经典微扰理论和哈密顿约简技术，提出了一种处理弱非线性波的通用方法。
The Hamiltonian description of hydrodynamic type systems in application to plasmas, hydrodynamics, and magnetohydrodynamics is reviewed with emphasis on the problem of introducing canonical variables. The relation to other Hamiltonian approaches, in particular natural-variable Poisson brackets, is pointed out. It is shown that the degeneracy of noncanonical Poisson brackets relates to a special type of symmetry, the relabeling transformations of fluid-particle Lagrangian markers, from which all known vorticity conservation theorems, such as Ertel's, Cauchy's, Kelvin's, as well as vorticity frozenness and the topological Hopf invariant, are derived. The application of canonical variables to collisionless plasma kinetics is described. The Hamiltonian structure of Benney's equations and of the Rossby wave equation is discussed. Davey–Stewartson's equation is given the Hamiltonian form. A general method for treating weakly nonlinear waves is presented based on classical perturbation theory and the Hamiltonian reduction technique.