Possible Forms of Generating Function

Given an arbitrary canonical transformationit may not be possible to treatandas independent variables since the condition for this to be possible is that the equationcan be solved to givein terms ofandsoso this transformation cannot be applied if there is some point for whichFor example, the identity transformation does not satisfy the required condition, nor does it cover the common case whereis a function of only:

Fortunately, alternatives exist. There is no reason to takeandalone as independent variables. We could for example, useandfor which we need to solvefor which we can do as long asAltogether there are four possible generating function, which, together with the associated conditions and transformations, are shown in the table below.