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.