Wikipedia and most authors denote four types of canonical transformations: $F_1(\mathbf{q},\mathbf{Q})$ , $F_2(\mathbf{q},\mathbf{P})$, $F_3(\mathbf{p},\mathbf{Q})$ and $F_4(\mathbf{p},\mathbf{P})$. However, is it possible (or necessary, perhaps) to define a transformation that mixes both "old" and "new" canonical variables?

For instance, the following transformation is canonical: $$ \begin{matrix} Q_1=q_1, & P_1=p_1-2p_2,\\ Q_2=p_2, & P_2=-2q_1-q_2. \end{matrix} $$

I have found that a possible generating function for this transformation would be $$ G(q_1,q_2,P_1,Q_2)=P_1q_1+Q_2(2q_1+q_2), $$ which doesn't fit any of the four traditional types.

P.S.: I found $G$ by (i) making a canonical tranformation $\tilde{Q}_2=-P_2$, $\tilde{P}_2=Q_2$, (ii) finding a $F_2$-type function and (iii) reverting back to $Q_2,P_2$.


Classical Mechanics (Third Edition), H. Goldstein, Chapter 9, Page 374:


Finally, note that a suitable generating function doesn't have to conform to one of the four basic types for all degrees of freedom of the system. It is possible. for some canonical transformations necessary, to use a generating function that is a mixture of the four types. To take a simple example, it may be desirable for a particular canonical transformation with two degrees of freedom to be defined by a generating function of the form $$F'(q_1,p_2,P_1,Q_2,t)$$[...]

I think that essentially answers your question.

  • $\begingroup$ @RenanNobuyukiHirayama I have practically adopted the view that if it is classical mechanics, it is in Goldstein! $\endgroup$ – Dvij Mankad Sep 17 '18 at 3:12

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