Canonical transformation between two given hamiltonians

AI Thread Summary
The discussion revolves around finding a canonical transformation (CT) that transforms the initial Hamiltonian H to the transformed Hamiltonian K. The user initially attempted to use the generating function method but encountered difficulties due to the separability of the solution, making it challenging to derive transformation equations. They clarified that the problem is not homework but a study exercise, seeking a general method for such transformations in Hamiltonian mechanics. Despite solving the specific problem using the Hamilton-Jacobi equation, they are still looking for a more straightforward approach to finding a CT between two given Hamiltonians. The conversation highlights the complexities involved in canonical transformations and the need for effective methods in Hamiltonian mechanics.
csco
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Hello everyone, I am given the inital hamiltonian H = (1/2)*(px2x4 - 2iypy + 1/x2) and the transformed hamiltonian K = (1/2)*(Px2 + Py2 + 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/∂Px, ∂F/∂Py and the px, py from H with ∂F/∂x, ∂F/∂y to get a partial differential equation for the function F(x, y, Px, Py, 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!
 
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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.
 
csco, The problem is, it sounds like homework. Assure us it is not and you may get some answers.
 
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?
 
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