Variational principles in continuum mechanics
Variational principles in continuum mechanics
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DOI：
10.1098/rspa.1968.0103
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发表时间：
1968-05
期刊：
影响因子：
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通讯作者：
R. Seliger;G. Whitham
中科院分区：
文献类型：
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作者：
R. Seliger;G. Whitham
本文讨论了流体动力学、等离子体动力学和弹性力学中变分原理问题的一般问题，即求给定方程组的变分原理。在连续介质力学中，当使用欧拉描述时，会出现困难；哈密顿原理的扩展在拉格朗日的描述中是直接的。我们发现，解决这些困难的方法是用Clebsch（1859）在等熵流体流动情况下引入的v =∇X + λ∇μ型表达式来表示欧拉速度v。林（1963）的工作阐明了与汉密尔顿原理的关系。本文还证明了电磁场的势表示和麦克斯韦方程组的变分原理可以拟合到同一个整体格式中。对水波、旋转和分层流体中的波、罗斯比波和等离子体波的方程给予了特别的关注，因为在最近的波传播工作中出现了对这些方程的变分公式的需要（Whitham 1967）。用“势表示法”（例如连续介质力学中的克莱施表示法和电磁学中的标量势和矢量势）求解一些方程，然后为其余方程找到变分原理，这似乎是解决一般问题的关键。用一个与Pfaff问题的微分形式的类比来支持这一观点。
Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇X + λ∇μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.