Is Any Dynamical Variable's Infinitesimal Transformation Canonical?

In summary, the problem at hand is to verify that the infinitesimal transformation generated by any dynamical variable g is a canonical transformation. The transformation can be represented as q_i' = q_i + δq_i and p_i' = p_i + δp_i, where δq_i and δp_i are infinitesimal changes. In order for the new coordinates to be canonical, the Poisson brackets between the new coordinates must satisfy the conditions: {q_j',q_k'} = 0 = {p_j',p_k'} and {q_j',p_k'} = δ_jk. The goal is to prove that these conditions are still satisfied under the given transformation.
  • #1
Lotophage
Problem: Verify that the infinitesimal transformation generated by any dynamical variable g is a canonical transformation.

I've worked out that an infinitesimal canonical transformation can be represented as follows:

q_i -> q_i' = q_i + ε(∂g/∂p_i) ≡q_i + δq_i

p_i -> p_i' = p_i - ε(∂g/∂q_i) ≡p_i + δp_i

I also know that, for any Hamiltonian, a new set of coordinates is only canonical if

{q_j',q_k'} = 0 = {p_j',p_k'}

==> {q_j',p_k'} = δ_jk

where {} are Poisson brackets and the lower scores denote subscript for the following character.

So my question is: Can I use these relationships to solve the problem? If so, how do I get started? The use of indices in these relationships confuses me, and I don't know if the δ_jk is supposed to be Kronecker delta, or some infinitesimal number. My intuition says that it's the Kronecker, but the indices used in the book I'm using are not explained, and I am unsure of how to apply the criterion for a canonical transformation to the general form of an infinitesimal transformation as given above. Any hints would be greatly appreciated!

Thanks,

Chris
 
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  • #2
Ok,

so the idea here is to prove that a certain type of transformation is canonical correct? So you need to think of what defines a canonical transformation and you have this already. The g in your equations is the quantity that is conserved if the transformation is canonical and the Hamiltonian is invariant under this transformation. You are correct that the term in your last equation is the kronecker delta. So the thing to do would be to check that your new coordinates, the ones with the primes, still obey your canonical transformation rules... that is the equations that define a canonical transformation are still satisfied.
Hope this helps. If this is unclear let me know.
Cheers
 
  • #3


To verify that the infinitesimal transformation generated by any dynamical variable g is a canonical transformation, we can use the definition of canonical transformation as given by the Hamilton equations:

{q_j,p_k} = δ_jk

where {} are Poisson brackets and the lower scores denote subscript for the following character.

Using the infinitesimal transformation given in the problem, we can rewrite the Hamilton equations as:

{q_j',p_k'} = {q_j + δq_j, p_k + δp_k} = δ_jk

Expanding the Poisson brackets, we get:

{q_j,p_k} + δp_k∂q_j/∂q_k - δq_j∂p_k/∂q_k = δ_jk

Since the first term is already equal to δ_jk, we can simplify the equation to:

δp_k∂q_j/∂q_k - δq_j∂p_k/∂q_k = 0

This shows that the infinitesimal transformation generated by g satisfies the Hamilton equations, and therefore it is a canonical transformation.

To further verify this, we can use the Jacobi identity for Poisson brackets:

{f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0

where f, g, and h are any dynamical variables.

Applying this identity to our infinitesimal transformation, we get:

{q_j,{p_k,g}} + {p_k,{g,q_j}} + {g,{q_j,p_k}} = 0

Substituting in the expressions for {q_j',p_k'} and expanding the Poisson brackets, we get:

{q_j,{p_k,g}} + {p_k,{g,q_j}} + {g,{q_j,p_k}} = δ_jk∂g/∂q_j - δ_jk∂g/∂p_k + δ_jk∂g/∂p_k = 0

Since δ_jk∂g/∂p_k = δ_jk∂g/∂q_j due to the symmetry of mixed partial derivatives, we can simplify the equation to:

2δ_jk∂g/∂p_k = 0

This shows that the infinitesimal transformation
 

1. What is a canonical transformation?

A canonical transformation is a transformation of coordinates and momenta in a physical system that preserves the Hamiltonian equations of motion. It is used to simplify the equations of motion and find new sets of coordinates that make the problem easier to solve.

2. What are the types of canonical transformations?

There are two types of canonical transformations: point transformations and generating function transformations. Point transformations involve a direct transformation of coordinates and momenta, while generating function transformations involve the use of a generating function to derive the new coordinates and momenta.

3. What is the importance of canonical transformations?

Canonical transformations are important in classical mechanics as they allow for the simplification of complicated systems and the identification of new conserved quantities. They also play a role in quantum mechanics and statistical mechanics.

4. How are canonical transformations related to symmetries?

Canonical transformations are closely related to symmetries in physics. A canonical transformation that preserves the Hamiltonian equations of motion is said to be a symmetry of the system, and vice versa. Symmetries allow for the identification of conserved quantities, which can simplify the equations of motion.

5. What are some applications of canonical transformations?

Canonical transformations have a wide range of applications in physics, including in classical mechanics, quantum mechanics, and statistical mechanics. They are used to simplify the equations of motion, identify conserved quantities, and solve complicated physical problems. They are also used in the study of symplectic geometry and in the formulation of Hamiltonian mechanics.

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