Canonical transformation between two given hamiltonians

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Discussion Overview

The discussion revolves around finding a canonical transformation (CT) between two given Hamiltonians, H and K. Participants explore methods for deriving such transformations, particularly in the context of Hamiltonian mechanics, and seek general approaches to tackle similar problems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant describes their initial Hamiltonian H and transformed Hamiltonian K, expressing difficulty in finding a canonical transformation between them.
  • The participant attempts to use the generating function of the second kind and derives a partial differential equation for the function F but struggles with the separability of the solution and the constants of integration.
  • Another participant questions whether the problem is homework-related, suggesting that clarification may lead to more responses.
  • A later reply clarifies that the problem is not homework but rather a study exercise, and the participant shares that they have solved the specific problem using the Hamilton-Jacobi equation, yet still seeks a general method for finding canonical transformations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific method for finding canonical transformations, and the discussion remains unresolved regarding general approaches to such problems.

Contextual Notes

The discussion highlights the challenges of dealing with constants of integration and the implications of separable solutions in the context of canonical transformations.

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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