Is There a Canonical Transformation for x = 2qa/sin(T) and p = 2qa.cos(T)?

In summary, the conversation discusses how to prove that a given transformation into new coordinates and momentum is canonical. The attempts include using matrix/jacobi method and symplectic method, but it is not clear if these methods are canonical. A clue is given to consider theorems that provide necessary and sufficient conditions for a transformation to be canonical. It is later mentioned that a generating function must be used, although the professor did not initially specify this. Some frustration is expressed towards the professor for not providing clear instructions.
  • #1
photomagnetic
13
0

Homework Statement


Show that x = 2qa/sin(T) and p = 2qa.cos(T) is a canonical transformation
into new coordinates T and momentum q.

Homework Equations

The Attempt at a Solution


It looks easy, I've tried matrix/jacobi method, and symplectic method. But these two seem to be not canonical. Am I missing something? The question doesn't give anything else. Do I have to find a generating function to prove that they are canonical? But then it'd would be silly, because the professor is very clear about these things. If he wanted to see a F generating function he would have said so.
 
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  • #2
Clue: do you have any theorems that provide necessary and sufficient conditions for a transformation to be canonical?
 
  • #3
I re-correct myself, the prof. is an jerk. He hadn't mentioned that we had to use a generating function. now I got it. thanks anyway.
 
  • #4
That's not nice. I'm sure your prof is a very nice fellow.

Incidentally, there are several different methods that you might have used to show that the given transformation is canonical.
 
  • #5
Yeah not telling which method we "must" use is not nice either.
Anyway after trying to do 3 hws each week, and spending a huge chunk of time,
people can get mad. Also no need to be politically correct here.
 

1. What are Canonical Transformations?

Canonical Transformations are mathematical operations used in classical mechanics to transform the coordinates and momenta of a system from one set of variables to another. They preserve the underlying structure of the system and are used to simplify the equations of motion.

2. Why are Canonical Transformations important?

Canonical Transformations are important because they allow us to simplify complex equations of motion and better understand the behavior of a system. They also help reveal symmetries and conservation laws in a system.

3. How do Canonical Transformations relate to Hamiltonian mechanics?

Canonical Transformations are closely related to Hamiltonian mechanics. They are used to transform the coordinates and momenta from Cartesian coordinates to the more general Hamiltonian coordinates, which are defined by the Hamiltonian function. This allows us to write the equations of motion in terms of the Hamiltonian and better understand the dynamics of a system.

4. What are the two types of Canonical Transformations?

The two types of Canonical Transformations are point transformations and generating function transformations. Point transformations preserve the form of the equations of motion, while generating function transformations preserve the Hamiltonian of the system.

5. How are Canonical Transformations performed?

Canonical Transformations are performed by using a set of transformation equations to convert the coordinates and momenta from one set of variables to another. These equations are derived from the fundamental Poisson bracket relations and the Hamiltonian function of the system.

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