# Canonical Transformation and harmonic-oscillator

• Shafikae
In summary, the transformation Q = p + iaq, P = (p-iaq)/2ia is canonical and the generating function can be obtained by solving for q in the original equation of Q = p + iaq and then taking the integral to solve for F. This transformation can be used to solve the harmonic oscillator problem by finding the Hamiltonian, which in this case is H = p^2/2m + (1/2)kq^2. The Poisson brackets are invariant, confirming the canonical nature of the transformation.
Shafikae
Show that the transformation
Q = p + iaq , P = (p-iaq)/2ia
is canonical and find the generating function. Use the transformation to solve the harmonic-oscillator problem.

I was able to determine if the transformation is canonical, and it is. However, when it came to finding the generating function, I wasnt getting it right. And how do we use this transformation to solve the harmonic oscillator?

Well, what do you know about generating functions?...What does the Hamiltonian become under this transformation?

I don't know how to obtain the Hamiltonian. But i know that to check whether the transformation is canonical, we can show that the Poisson brackets are invariant. Which I have shown they are. I'm not getting the right generating function, or is the generating function really the hamiltonian??

Shafikae said:
I'm not getting the right generating function, or is the generating function really the hamiltonian??

Well, what are you doing to obtain the (incorrect) generating function?

I don't know how to obtain the Hamiltonian.

You don't know what the Hamiltonian of the one-dimensional harmonic oscillator is?

Ok so I take the poisson bracket and obtain 1, therefore its canonical. I have an example similar to this, so i take q = - partial F / partial p , so i solve from the original equation of Q = p + iaq, i solve for q and then take the integral and solve for F.

Would the hamiltonian of a harmonic oscillator be H = p2/2 +(1/2)kx2

sorry i meanH = p2/2m +(1/2)kx2

H = p2/2m +(1/2)kq2

## 1. What is a canonical transformation?

A canonical transformation is a change of variables in the phase space of a physical system that preserves the Hamiltonian equations of motion. It is a mathematical tool used to simplify the equations of motion and reveal underlying symmetries in a system.

## 2. How does a canonical transformation relate to the harmonic oscillator?

The harmonic oscillator is a commonly studied physical system in which a particle's motion is governed by a restoring force that is proportional to the displacement from equilibrium. A canonical transformation can be used to transform the equations of motion for the harmonic oscillator into a simpler form, making it easier to analyze and solve.

## 3. What is the importance of Poisson brackets in canonical transformations?

Poisson brackets play a crucial role in canonical transformations as they are used to define the new coordinates and momenta in the transformed phase space. They also help to ensure that the transformed equations of motion remain consistent with the Hamiltonian equations.

## 4. Can a canonical transformation be used to transform any physical system?

Yes, a canonical transformation can be applied to any physical system as long as it satisfies the criteria for a symplectic transformation. This means that the transformation must preserve the symplectic structure of the system, which is closely related to the conservation of energy and momentum.

## 5. How is a canonical transformation different from a change of variables in classical mechanics?

A canonical transformation is a specific type of change of variables in classical mechanics that preserves the Hamiltonian equations of motion. This means that the transformed equations of motion will have the same solutions as the original equations. In contrast, a general change of variables can alter the solutions of the equations of motion.

Replies
3
Views
935
Replies
1
Views
808
Replies
16
Views
745
Replies
2
Views
1K
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
1K
• Classical Physics
Replies
4
Views
768
Replies
19
Views
2K
• Classical Physics
Replies
3
Views
681
• Mechanics
Replies
4
Views
811