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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.

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.

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.

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user87745
user87745

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.