认识勒让德变换  

Understanding the Legendre transform

作　　者：阎玉立[1] 张印杰[1] 

机构地区：[1]河北大学物理科学与技术学院,河北保定071002

出　　处：《大学物理》2013年第11期8-10,32,共4页College Physics

基　　金：国家自然科学基金资助项目(10947167);河北大学启动基金(2007-113)

摘　　要：勒让德变换是理论物理学中一个重要的工具,在经典力学、统计力学和热力学中起到重要的作用.然而,在本科生的课程中,它变化的灵活性和形式的优雅性常常被忽视,不像傅里叶变换那样经常被应用.本文从数学形式上修改了多变量函数的勒让德变换的表述方式,得到明显的具有对称性的方程,给出了勒让德变换与偏微商的关系,然后把结果应用到经典力学,推导出正则方程,应用到热力学中,导出热力学中重要的几个热力学函数及麦氏关系等.The Legendre transform is an important tool in theoretical physics, which plays a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in undergraduate courses, the power of mo- tivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by the Fourier trans- form. We modify the formal mathematics of the Legendre transform of multivariate function, resulting in manifestly symmetric equations, deriving the relation between the Legendre transform and the partial differentiation. Then we bring the results in the classical mechanics, derive canonical equation, and bring the results in the thermodynamic, as well as thermodynamic functions and Maxwell relation are also derived.

关 键 词：勒让德变换 对称 经典力学 热力学 

分 类 号：O414.2[理学—理论物理]