HAMILTONIAN FLUID MECHANICS
HAMILTONIAN FLUID MECHANICS
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DOI：
10.1146/annurev.fl.20.010188.001301
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发表时间：
1988
影响因子：
27.7
通讯作者：
R. Salmon
中科院分区：
文献类型：
--
作者：
R. Salmon
本文评述了哈密顿力学方法在流体力学中的应用。我所说的哈密顿力学是指所有常被称为经典力学的东西--这是Lanczos（1970）、Goldstein（1980）和Arnol'd（1978）的教科书的主题。自量子力学出现以来，哈密顿方法在粒子和场的经典和量子力学中扮演着越来越重要的角色。相比之下，将哈密顿方法引入流体力学的工作则比较缓慢。为什么会这样呢？在一般的力学系统中，拉格朗日或哈密顿运动方程是控制有质量的质点或刚体的位置和速度的耦合方程。这些耦合方程通常不能在没有找到所有其他因变量的情况下针对因变量的任何子集求解。相比之下，传统的欧拉流体方程是速度、密度和熵（将压力视为密度和熵的规定函数）的封闭方程，可以（原则上）在不找到每个流体粒子的轨迹的情况下求解。一旦速度场已知，粒子的输运总是可以通过求解三个独立的被动平流示踪剂（如初始笛卡尔分量）的方程来重建，但如果只寻求欧拉场，则不需要这些额外的计算。在等密度流动的特殊情况下，欧拉方程比一般的拉格朗日方程或哈米尔牛顿方程简单得多。从哈密尔顿的观点来看，欧拉描述的异常简单来自于流体的对称性
This paper reviews the relatively recent application of the methods of Hamiltonian mechanics to problems in fluid dynamics. By Hamiltonian mechanics I mean all of what is often called classical mechanics-the subject of the textbooks by Lanczos ( 1970), Goldstein ( 1 980), and Arnol'd (1978). Since the advent of quantum mechanics, Hamiltonian methods have played an increasingly important role in both the classical and quan tum mechanics of particles and fields. By comparison, the introduction of Hamiltonian methods into fluid mechanics has been tardy. Why is this so? In general mechanical systems, the Lagrangian or Hamiltonian equa tions of motion are coupled equations governing the locations and veloc ities of massive particles or rigid bodies. These coupled equations cannot generally be solved for any subset of the dependent variables without also finding all of the other dependent variables. By contrast, the conventional Eulerian fluid equations are closed equations in the velocity, density, and entropy (regarding pressure as a prescribed function of the density and entropy) that can (in principle) be solved without also finding the trajectory of every fluid particle. Once the velocity field is known, the particle tra jectories can always be reconstructed by solving the equations for three independent, passively advected tracers (such as the initial Cartesian com ponents), but these extra computations are not required if only the Eulerian fields are sought. In the special case of constant-density flow, the Eulerian equations are dramatically simpler than the general Lagrangian or Hamil tonian equations for the fluid. From the Hamiltonian perspective, the extraordinary simplicity of the Eulerian description derives from a symmetry property of the fluid