Verifying a Canonical Transformation with Poisson Brackets

In summary, a canonical transformation is a change of coordinates in a Hamiltonian system that preserves the form of Hamilton's equations and the conserved quantities. Poisson brackets are used to verify if a transformation is canonical, and the transformation must satisfy two properties: preserving Hamilton's equations and the Poisson brackets between canonical variables. To verify a transformation, the Poisson brackets of the original and new coordinates and momenta must be equal. It is important to verify a canonical transformation to ensure the conservation of important quantities and to use Hamiltonian mechanics to analyze and solve the system.
  • #1
darida
37
1

Homework Statement



Show that

[itex]Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω})[/itex]
[itex]Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω})[/itex]
[itex]P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2})[/itex]
[itex]P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2})[/itex]

(where mω is a constant) is a canonical transformation by Poisson bracket test. This requires evaluating six simple Poisson brackets.

2. The attempt at a solution

[itex][Q_{1},P_{1}]=[\frac{∂Q_{1}}{∂q_{1}}\frac{∂P_{1}}{∂p_{1}}-\frac{∂Q_{1}}{∂p_{1}}\frac{∂P_{1}}{∂q_{1}}]+[\frac{∂Q_{1}}{∂q_{2}}\frac{∂P_{1}}{∂p_{2}}-\frac{∂Q_{1}}{∂p_{2}}\frac{∂P_{1}}{∂q_{2}}][/itex]
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.
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etc

Correct?
 
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  • #3
Thank you!
 

1. What is a canonical transformation?

A canonical transformation is a change of coordinates in a Hamiltonian system that preserves the form of Hamilton's equations, ensuring that the equations of motion and the conserved quantities remain unchanged.

2. How do Poisson brackets relate to canonical transformations?

Poisson brackets are used to verify if a transformation is canonical by checking if they remain unchanged under the transformation. If the Poisson brackets of the original coordinates and momenta are equal to the Poisson brackets of the new coordinates and momenta, then the transformation is canonical.

3. What are the properties of a canonical transformation?

A canonical transformation must satisfy two properties: (1) it must preserve the form of Hamilton's equations, and (2) it must preserve the Poisson brackets between the canonical variables.

4. How do you verify a canonical transformation with Poisson brackets?

To verify that a transformation is canonical using Poisson brackets, you need to calculate the Poisson brackets of the original coordinates and momenta and compare them to the Poisson brackets of the new coordinates and momenta. If they are equal, then the transformation is canonical.

5. Why is it important to verify that a transformation is canonical?

Verifying that a transformation is canonical is important because it ensures the conservation of important quantities, such as energy and momentum, in a Hamiltonian system. It also allows us to use the powerful tools of Hamiltonian mechanics, such as the Hamiltonian function, to analyze and solve the system.

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