拉格朗日理论是理论力学和天体力学重要的教学内容. 理想流体作为场论的典范,也是理论力学、天体力学和场论重要的教学内容. 但有趣的是从拉格朗日理论的角度来讲解流体力学的教科书并不多. 本文把热力学、广义坐标变换和微分同胚变换结合到一起来分析理想流体的拉格朗日理论. 本文特别讲述了一个关于拉格朗日理论中可以作为广义坐标的限制性数学定理. 此定理表明了只用粒子数密度、流体能量密度和四速场这三个场量作为广义坐标的理想流体拉氏场论不存在. 本文的论述不仅可以帮助读者深入理解流体力学中的热力学、广义相对论中的微分同胚变换和拉格朗日理论中的广义坐标变换以及关于流体的拉格朗日理论描述, 同时还可以帮助读者进一步深入理解拉格朗日理论本身. 这些内容可以作为理论力学、天体力学、场论和相对论课堂教学的有益补充.

Xing ZiZheng1, He XiaoKai2, CAO ZhouJian1
(1. School of Physics and Astronomy,Beijing Normal University, Beijing 100875, China; 
2. School of Mathematics and Statistics, Hunan First Normal University, Changsha, Hunan 410205, China)

Abstract:Lagrange theory is an important teaching content in theoretical mechanics and celestial mechanics. The perfect fluid, as a paradigm of field theory, is also an important teaching content in theoretical mechanics, celestial mechanics, and field theory. But interestingly, there are few textbooks that explain fluid mechanics from the perspective of Lagrangian theory. This article combines thermodynamics, generalized coordinate transformation, and differomorphism together to analyze the Lagrangian theory of perfect fluids.  A mathematical theorem which provides a restriction condition for the generalized coordinates of Lagrange theory is specifically discussed. This theorem demonstrates the non-existence of a Lagrangian theory for a perfect fluid with only three field quantities including particle number density, fluid energy density, and four velocity field as generalized coordinates. The content of this article can not only help readers deepen their understanding on thermodynamics in fluid mechanics, differomorphism transformation in general relativity, generalized coordinate transformation in Lagrangian theory, and Lagrangian theory description of fluids, but also help readers further understand Lagrangian theory itself. The contents