Legendre变换的意义
Making Sense of the Legendre Transform
摘要：
勒让德变换是理论物理中的一个重要工具，在经典力学、统计力学和热力学中起着关键作用。然而，在典型的本科或研究生课程中，动力和优雅的方法往往是失踪，不像傅立叶变换经常享受的待遇。我们审查和修改的方式，阐明了正式的数学，导致明显的对称方程的勒让德变换的介绍，从而澄清代数和几何结构的变换。然后，我们引入物理学来激发转换，作为选择更容易控制的独立变量的一种方式。我们演示了如何自然产生的勒让德变换从统计力学，并显示如何使用无量纲热力学势导致更自然和对称的关系。
英文摘要：
The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.
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