Canonical Transformations

In summary: Actually the definition of canonical transformation usually stipulates that λ=1 in the above requirement. For any other λ, we call these "extended canonical transformations".This is very general because otherwise the action would not remain stationary for the actual motion. This is because the freedom in the Lagrangian is only in the total time derivative term.
  • #1
A_B
93
1
Hi,

Im working through some chapters of Goldstein and I'm up to canonical transformations now. On page 370 it says that the variational principle for the hamiltonians K and H are both satisfied if H and K are connected by a relation of the form

λ(pq' - H) = PQ' - K + dF/dt

And I can see this. My question is, are there canonical transformations that do not fit this relation? And if so are they impportant? Is this relation a very general one, or does it simply turn out to give good tranformations in many problems?

A_B
 
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  • #2
Also if we obtain a relation

pq' - H = PQ' - K + ∂F/∂t + (∂F/∂q)q' + (∂F/∂Q)Q'

Goldstein says (p 372 eq 9.13)

"Since the old and the new coordinates, q and Q, ere seperately independent, the above equation can hold identically only if the coefficients of q' and Q' each vanish:"

leading to

p = ∂F/∂q,

P = -∂F/∂Q

I don't see why this is necessary for the relation to hold.
 
  • #3
A_B said:
Hi,

Im working through some chapters of Goldstein and I'm up to canonical transformations now. On page 370 it says that the variational principle for the hamiltonians K and H are both satisfied if H and K are connected by a relation of the form

λ(pq' - H) = PQ' - K + dF/dt

And I can see this. My question is, are there canonical transformations that do not fit this relation? And if so are they impportant? Is this relation a very general one, or does it simply turn out to give good tranformations in many problems?

A_B

Actually the definition of canonical transformation usually stipulates that λ=1 in the above requirement. For any other λ, we call these "extended canonical transformations".

This is very general because otherwise the action would not remain stationary for the actual motion. This is because the freedom in the Lagrangian is only in the total time derivative term.
 
  • #4
can someone answer my question in the second post?

thanks!
A_B
 
  • #5



Hello A_B,

Thank you for your question about canonical transformations. The relation you mentioned, λ(pq' - H) = PQ' - K + dF/dt, is known as the generating function for canonical transformations. This relation is indeed a very general one and it is satisfied for most canonical transformations. However, there are some cases where this relation does not hold and these transformations are known as non-canonical transformations.

Non-canonical transformations are still important in certain problems, especially in cases where the Hamiltonian is not explicitly time-dependent. In these cases, the generating function relation does not hold and different methods must be used to find the canonical transformation. Non-canonical transformations also have applications in quantum mechanics and statistical mechanics.

Overall, the generating function relation is a powerful tool for finding canonical transformations in many problems, but it is not the only way to do so. I hope this helps to clarify your question. Keep exploring and learning about canonical transformations, they are an important concept in classical mechanics. Good luck with your studies!

Best,
 

1. What are Canonical Transformations?

Canonical transformations are mathematical transformations that preserve the fundamental structure of a system, such as its equations of motion or its symmetries. They are used in classical mechanics and quantum mechanics to simplify the equations of motion and find new coordinates that make solving the equations easier.

2. Why are Canonical Transformations important?

Canonical transformations are important because they allow us to simplify and solve complex problems in physics, such as the motion of planets in a solar system or the behavior of particles in a quantum system. They also help us better understand the underlying symmetries and structure of a system.

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

A canonical transformation is a specific type of coordinate transformation that preserves the fundamental structure of a system, while a general coordinate transformation can change the structure of a system. In other words, a canonical transformation is a more restricted type of coordinate transformation with specific properties.

4. How do you determine if a transformation is canonical?

To determine if a transformation is canonical, we use a set of equations called the canonical equations, which describe the transformation from the old coordinates to the new coordinates. If the canonical equations hold, then the transformation is canonical.

5. What are some common examples of Canonical Transformations?

Some common examples of canonical transformations include the transformation from Cartesian coordinates to polar coordinates, the transformation from position and momentum coordinates to action-angle coordinates, and the transformation from the Hamiltonian formalism to the Lagrangian formalism. These transformations are commonly used in classical mechanics and quantum mechanics to simplify and solve problems.

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