Abstract
The mutual relationships between four generating functionsF1(q, Q), F2(q, P), F3(p, P), F4(p, Q) and four kinds of canonical variables q. p. Q, P concerned in Hamilton'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 homomourphic to D4 point group of 8 elements with correspondence of 4:1. Transformations and relationships of four state functionsG(P,T), H(P,S), U(V,S), F(V,T) and four variables P, V, T. S in thermodynamics, are just the some Lagendre's transformations with the relationships of canonical transformations. The state functions of thermodynamics are summarilty 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.
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