Confusion about canonical transformations

[jojo12345]

I have read that "any" function of new and old coordinates may be used to generate a canonical transformation. However, it seems there must be some restriction on the generating function.

For example, if you try to define a generating function for a system with 2 degrees of freedom:

G(x,y,X,Y) = sin(X+Y) + sin(x+y)

where x,y are the old coordinates and X,Y the new, and take the appropriate partial derivatives, you seem to end up with:

px = cos(x+y)
py = cos(x+y)

Px = cos(X+Y)
Py = cos(X+Y) (Px,Py are the new momenta and px,py the old)

Not only does this seem to tell you nothing about a "transformation", but it seems to imply px = py, Py = Px.

What is going on here?

 

[Valerie Park]

Is there a general restriction on the generating function for a canonical transformation?Yes, there is a general restriction on the generating function for a canonical transformation. The generating function must be a function of both the new and old coordinates that is single valued and has a continuous first partial derivative. Furthermore, the generating function must satisfy the Hamilton-Jacobi equation. This ensures that the transformation is canonical.

 

[charng]

There seems to be a misunderstanding about canonical transformations. While it is true that any function of old and new coordinates can be used to generate a canonical transformation, there are certain restrictions on the generating function in order for it to be considered a canonical transformation.

Firstly, the generating function must be a function of both the old and new coordinates, as well as the old and new momenta. In the example given, the generating function only depends on the old and new coordinates, and not the momenta. This is why the resulting equations for the momenta are the same for both the old and new coordinates, leading to the confusion of px = py and Px = Py.

Additionally, the generating function must be a smooth and invertible function, meaning that it must have a unique inverse that can be solved for the old coordinates and momenta in terms of the new coordinates and momenta. This is important in order for the transformation to be reversible and for the Hamiltonian equations of motion to be preserved.

In summary, while it is possible to use any function of old and new coordinates to generate a canonical transformation, it must also satisfy the above restrictions in order to be considered a valid transformation. It is important to carefully check the equations and ensure that all variables are accounted for in order to avoid the confusion seen in the example given.