正则变换的一种群表示

经典力学中的哈密顿正则变换所涉及的4个母函数F₁(q,Q),F₂(q,P),F₃(p,P),F₄(p,Q)和4种正则变量q,p,Q,P之间所有的关系,可以由7个基本关系式经线性变换而得到,这些变换是勒让德变换,变换是由32个8×8的变换矩阵来实现的,而这32个矩阵以4:1的关系与具有8个群元的D₄点群同态。热力学中的4个状态函数G(P,T),H(P,S),U(V,S),F(V,T)和4个热力学变量P,V,T,S之间的变换关系恰好与正则变换关系相同。热力学状态方程是源于宏观测量的实验结果的概括,而哈密顿正则变换是经典力学的理论性总结,它们的群表示是相同的,即它们的数学结构是相同的, 这种共性表明热力学变换是一维哈密顿正则变换的实例。

A Group Representation of Canonical Transformation

The mutual relationships between four generating functions F₁(q,Q),F₂(q,P),F₃(p,P),F₄(p,Q) and four kinds of canonical variables q,p,Q,P concerned in Hamiltion's canonical transformations,can be got with linear transformations from seven basic formulae.All of them are Legendre's transformation which are implemented by 32 matrices of 8×8 which are homomorphic to D₄ point group of 8 elements with correspondence of 4:1.Transformations and relationships of four state functions G(P,T),H(P,S),U(V,S),F(V,T) and four variables P,V,T,S in thermodynamics,are just the same Lagendre's transformations with the relationships of canonical transformations.The state functions of thermodynamics are summarily founded on experimental results of macroscrope measurements,and Hamilton's canonical transformations are theoretical generalization of classical mechanics,Both group represents are the same,and it is to say,their mathematical frames are the same.This generality indicates the thermodynamical transformation is an example of one-dimensional Hamilton's canonical transformation.