A Most General form of Canonical Transformation

[kolawoletech]

How do I go about finding the most general form of the canonical transformation of the form
Q = f(q) + g(p)
P = c[f(q) + h(p)]
where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and conjugate momentum in the new and old system

 

[vanhees71]

Isn't this "homework" of some kind? You should post it in the Homework and Coursework section! Anyway, here's some hint:

I'd try to determine constraints on the functions by evaluating the Poisson brackets which must be
$$\{Q,Q\}=\{P,P \}=0, \quad \{Q,P \}=1$$
in order to have a canonical transformation.

 

[kolawoletech]

I am done all that with certain kinds of relationship between (Q,P) and (q,p) but I am unable to do so with this general formula that does not give the function itself