The aim is, of course, to find a new system that is easier to solve. Pure coordinate transformations can do a little bit of this, for example a two body system is equivalent to a one-body system with an effective mass if we transform to center of mass coordinates. If the potential has a symmetry, e.g. does not depend on the position of the center of mass, then the effecive one body system is much easier to solve. In this case, both systems are still closely related. The nature of the force is the same in both cases.
Canonical transformations are much more powerful, they have much larger effects.
We make use of a so-called generating function . A generating function depends on a set of old coordinates and a set of new coordinates, and possibly time. In our example we use The one label on F simply indicates that this is choice one. For the other three possibilities this varies. Look at the book for the other three definitions.