Canonical transformation between two given hamiltonians

In summary, Bill attempted to solve a problem of finding a canonical transformation between two hamiltonians, but was unable to do so because the solution was separable. He also mentioned that his method of solving the Hamilton-Jacobi equation twice may be lengthy, so his question is still the same.
  • #1
csco
14
0
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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  • #2
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.
 
  • #3
csco, The problem is, it sounds like homework. Assure us it is not and you may get some answers.
 
  • #4
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?
 
  • #5


Dear fellow scientist,

Thank you for sharing your problem with us. The process of finding a canonical transformation between two given Hamiltonians can be challenging, but it is an important skill to have in the field of physics. I will try to provide some guidance and suggestions on how to approach this problem.

First, it is important to understand the concept of a canonical transformation. A canonical transformation is a transformation in phase space that preserves the Hamilton's equations of motion. In other words, the transformed Hamiltonian should have the same form as the initial Hamiltonian, only with different variables. In your case, we have two different Hamiltonians, H and K, and we need to find a transformation that will make them equivalent.

One approach to solving this problem is to use the generating function method, as you have attempted. However, it is important to note that there are different types of generating functions, and in this case, you need to use the generating function of the second kind, also known as the extended Hamilton-Jacobi equation. This type of generating function has the form F = F(q,Q,t), where q and Q are the old and new coordinates, respectively. In your case, q = (x,y) and Q = (Px,Py). This function can be used to express the new Hamiltonian K in terms of the old Hamiltonian H and the old and new coordinates.

Another approach is to use the transformation equations directly. In this case, you would need to find a set of equations that relate the old and new coordinates and momenta. These equations can be found by equating the Hamilton's equations for H and K. This method may be more straightforward, but it requires some algebraic manipulation and can be time-consuming.

It is also important to note that the solution to the canonical transformation may not always be unique. In some cases, there may be multiple transformations that can yield the same result. Therefore, it is always a good idea to check your solution and make sure it satisfies the necessary conditions of a canonical transformation.

I hope this helps you in your problem-solving process. If you need further assistance, I suggest consulting with a colleague or a mentor who has experience with canonical transformations. Good luck!
 

What is a canonical transformation?

A canonical transformation is a change of coordinates in phase space that preserves the Hamiltonian equations of motion. It is a useful tool in classical mechanics for simplifying the equations of motion and finding new conserved quantities.

What are the conditions for a transformation to be canonical?

There are two conditions for a transformation to be canonical: the original Hamiltonian equations of motion must still hold in the new coordinates, and the transformation must preserve the Poisson bracket structure of the original coordinates.

How are canonical transformations related to symmetries?

Canonical transformations are closely related to symmetries in classical mechanics. In fact, every symmetry in the Hamiltonian leads to a specific type of canonical transformation. This allows for an elegant way to find conserved quantities and simplify equations of motion.

What is the difference between a canonical transformation and a coordinate transformation?

A canonical transformation only changes the coordinates in phase space, while the Hamiltonian and equations of motion remain the same. A coordinate transformation, on the other hand, changes the coordinates in both phase space and configuration space, and can change the Hamiltonian and equations of motion.

How can I determine if a given transformation is canonical?

To determine if a transformation is canonical, you can check if the Poisson bracket structure is preserved and if the equations of motion still hold in the new coordinates. Additionally, there are specific conditions and techniques for determining canonical transformations for certain symmetries or types of Hamiltonians.

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