Canonical transformation between two given hamiltonians

[csco]

Hello everyone, I am given the inital hamiltonian H = (1/2)*(p_x²x⁴ - 2iyp_y + 1/x²) and the transformed hamiltonian K = (1/2)*(P_x² + P_y² + X² + Y²) and I'm supposed to show there exists a canonical transformation that transforms H to K and find it. I don't know how to solve problems of this sort. I can find the canonical transformation given the generating function so I started with the equation K = H + ∂F/∂t where F is the generating function of the second kind and replaced the X, Y from K with ∂F/∂P_x, ∂F/∂P_y and the p_x, p_y from H with ∂F/∂x, ∂F/∂y to get a partial differential equation for the function F(x, y, P_x, P_y, t). Assuming a separable solution I solve the equation and obtain a generating function F but it's impossible to get transformation equations from this F because the solution is separable. Also I can't make sense of the constants of integrations in this situation.

How does one deal with these kind of problems?
Any help is appreciated!

 

[csco]

No answers? I thought finding a CT between two given hamiltonians would be a standard problem that I just didn't know how to solve.

 

[Bill_K]

csco, The problem is, it sounds like homework. Assure us it is not and you may get some answers.

 

[csco]

Bill: it isn't homework. I found this problem while studying and it made me rethink about a few things in hamiltonian mechanics. I only wanted to know of a general method to solve problems of this sort since I had never seen them before that's why the question on how to deal with them. I gave details on the problem simply because what seemed to me the most obvious method to obtain a solution didn't give me a solution that made sense and I was wondering if there was a reason for this.
I solved that specific problem already but my method to find a CT between H and K is solving the Hamilton-Jacobi equation twice to do H -> 0 -> K which is rather lengthy. So my question is still the same, how does one deal with the problem of finding a CT given H and K?