How Do Poisson Brackets Show Non-Canonical Transformations in Magnetic Fields?

In summary: Therefore, for i = 1, the Poisson bracket is equal to 1. You can repeat this calculation for i = 2 and i = 3 to show that the Poisson bracket is also equal to 1 for these cases.In summary, by calculating the Poisson brackets for each component separately, we can see that the transformation given in the problem is indeed canonical.
  • #1
mc0210
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Homework Statement


Consider the motion of a particle with charge e in a homogenous magnetic field B_i. The Hamiltonian for this problem is $$H = \frac{1}{2m} \sum_{i=1}^{i=3} \left[ p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k\right]^2.$$ By calculating the Poisson brackets, show that the transformation $$p_i \rightarrow p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k$$ is not canonical.

Homework Equations


None that aren't already given.

3. Attempt at Solution
I feel decently comfortable using Poisson brackets, however I am not sure how to perform this calculation because of the summation from i=1 to i=3. My attempt was to add each component, so start with x_i and p_i and perform the poisson bracket and get 1, then repeat for x_j and p_j, and for x_k and p_k. Since each yields one, I got a total of 3. To be canonical it must equal 1. However, I really don't think this method is correct. Thanks for any help!
 
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  • #2

Thank you for your question. I am happy to help you with this problem.

Firstly, let's review the definition of a canonical transformation. A canonical transformation is a transformation of coordinates and momenta that preserves the form of Hamilton's equations. In other words, the Poisson bracket of the new coordinates and momenta should equal 1, as you correctly stated.

Now, let's look at the transformation given in the problem: $$p_i \rightarrow p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k$$
We can rewrite this transformation as: $$p_i \rightarrow p_i + \frac{e}{2}\epsilon _{ijk}x_kB_j$$
Note that this transformation involves both the coordinates and momenta, which means we need to consider both in our calculation of the Poisson bracket.

As you correctly stated, we need to calculate the Poisson bracket for each component (i,j,k) separately. Let's start with i = 1:
$$\{x_1, p_1 + \frac{e}{2}\epsilon _{ijk}x_kB_j\} = \frac{\partial x_1}{\partial x_1}\frac{\partial p_1}{\partial p_1} + \frac{\partial x_1}{\partial p_1}\frac{\partial p_1}{\partial x_1} + \frac{\partial x_1}{\partial x_2}\frac{\partial p_1}{\partial p_2} + \frac{\partial x_1}{\partial p_2}\frac{\partial p_1}{\partial x_2} + \frac{\partial x_1}{\partial x_3}\frac{\partial p_1}{\partial p_3} + \frac{\partial x_1}{\partial p_3}\frac{\partial p_1}{\partial x_3}$$
Note that we only need to consider the terms that involve p_1 and x_1, as all other terms will evaluate to 0 due to the Kronecker delta function. Also, since the coordinates and momenta are independent, we can set the partial derivatives equal to 0. This gives us:
$$\{x_1, p_1 + \frac{e}{2}\epsilon _{ijk}x_kB_j
 

Related to How Do Poisson Brackets Show Non-Canonical Transformations in Magnetic Fields?

What are Poisson brackets and how are they used in science?

Poisson brackets are a mathematical tool used in classical mechanics to describe the dynamics of a physical system. They are used to calculate the time evolution of a system's properties, such as position and momentum.

How do you calculate Poisson brackets?

Poisson brackets are calculated using the formula {A,B} = ∑(∂A/∂qi)(∂B/∂pi) - (∂A/∂pi)(∂B/∂qi), where A and B are the two properties being evaluated, and qi and pi are the general coordinates and momenta of the system.

What is the significance of Poisson brackets in Hamiltonian mechanics?

Poisson brackets play a crucial role in Hamiltonian mechanics, as they provide a way to express Hamilton's equations of motion, which describe the evolution of a system over time. They also serve as a way to determine the conserved quantities of a system, such as energy and momentum.

Can Poisson brackets be used in other areas of science besides classical mechanics?

While Poisson brackets were originally developed for use in classical mechanics, they have also been applied in other areas of physics, such as quantum mechanics and statistical mechanics. They have also found applications in fields such as economics, biology, and engineering.

What are the limitations of using Poisson brackets?

Poisson brackets are only applicable to systems that follow classical mechanics principles, and cannot be used to describe quantum systems. Additionally, they may not accurately describe systems with strong nonlinearities or chaotic behavior. In these cases, other mathematical tools may be more appropriate.

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