Given a canonical transformation, how does one find its type?

In summary, the given transformation involves four coordinates: x, y, p_x, and p_y. To determine the type of transformation, one must find a generating function that would allow for the other coordinates (X, Y, P_X, and P_Y) to be written in terms of these four. Starting from Hamilton's principle and trying the definitions of each type of generator is not necessary. Instead, one can try various generating functions and see if they lead to a coordinate set that works.
  • #1
mjordan2nd
177
1
I'm given the following transformation

[tex]X=x \cos \alpha - \frac{p_y}{\beta} \sin \alpha[/tex]
[tex]Y=y \cos \alpha - \frac{p_x}{\beta} \sin \alpha[/tex]
[tex]P_X=\beta y \sin \alpha + p_x \cos \alpha[/tex]
[tex]P_Y=\beta x \sin \alpha + p_y \cos \alpha[/tex]

and I'm asked to find what type(s) of transformation it is. I'm not sure how to go about doing this without being given a generator. Do I basically try and find a generator? If so, how do I do this. Do I start from Hamilton's principle and try the definition of each type of generator?
 
Physics news on Phys.org
  • #2
Do I basically try and find a generator?

Not quite, that is a lot more work. You don't have to actually find the generator to determine what type it should be.

If so, how do I do this. Do I start from Hamilton's principle and try the definition of each type of generator?

Say, for the sake of argument, the generator [itex]F[/itex] is a function of [itex]p_x,y,X,Y[/itex] (it's not) so that [itex]F=F(p_x,y,X,Y)[/itex]. This means that you would be able to write the other coordinates (namely [itex]p_y,x,P_x,P_y[/itex]) in terms of these ones. I can see that this type of generating function won't work since when I try to write [itex]P_y[/itex] I find (you'll need to invert the transformation to see this):
[tex]
P_y = x\beta \sin\alpha + p_y\cos\alpha = \left(X\cos\alpha + \frac{\sin\alpha}{\beta}P_y\right) \beta\sin\alpha + p_y\cos\alpha
[/tex]
and that simplifies to [itex]P_y=X\beta\tan\alpha + p_y\sec\alpha [/itex]. Therefore, I cannot write [itex]P_y[/itex] as a function of only those chosen coordinates. Try something else and see if you can find a coordinate set that works---then you will know what type of generating function you can have.
 

1. What is a canonical transformation?

A canonical transformation is a mathematical transformation that preserves the fundamental structure and equations of a physical system. It is a change of coordinates that maintains the Hamiltonian form of the equations of motion.

2. How is a canonical transformation different from other transformations?

A canonical transformation is different from other transformations because it preserves the symplectic structure of a system, meaning that it conserves the volume of the phase space. This is not the case for other types of transformations, such as linear or nonlinear transformations.

3. What are the types of canonical transformations?

There are three types of canonical transformations: symplectic, contact, and Poisson. Symplectic transformations preserve the symplectic structure and are reversible, while contact transformations preserve the Hamiltonian form but may not be reversible. Poisson transformations preserve the Poisson brackets, which describe the fundamental relationships between physical quantities.

4. How does one determine the type of a given canonical transformation?

The type of a given canonical transformation can be determined by examining its effect on the symplectic structure, Hamiltonian form, and Poisson brackets. For example, a transformation that preserves all three is a symplectic transformation, while a transformation that only preserves the Poisson brackets is a Poisson transformation.

5. What are some applications of canonical transformations?

Canonical transformations have various applications in physics and engineering, such as in the study of classical mechanics, Hamiltonian dynamics, and statistical mechanics. They are also used in the analysis of electromagnetic systems, quantum mechanics, and control theory.

Similar threads

  • Classical Physics
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
579
Replies
6
Views
909
  • Advanced Physics Homework Help
Replies
4
Views
267
  • Introductory Physics Homework Help
Replies
29
Views
775
  • Advanced Physics Homework Help
Replies
19
Views
1K
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
643
  • Classical Physics
Replies
7
Views
633
  • Electrical Engineering
Replies
1
Views
956
Back
Top