Poisson brackets for a particle in a magnetic field

[joriarty]

I'm struggling to understand Poisson brackets a little here... excerpt from some notes:

We’ve seen in the example of section 4.1.3 that a particle in a magnetic field $$\textbf{B} = ∇×\textbf{A}$$
is described by the Hamiltonian
$$H = \frac{1}{2m}\left( \textbf{p} - \frac{e}{c} \textbf{a} (\textbf{r} ) \right)^2 = \frac{m}{2} \dot{\textbf{r}}^2$$
where, as usual in the Hamiltonian, $$\dot{\textbf{r}}$$ is to be thought of as a function of r and p. It’s a simple matter to compute the Poisson bracket structure for this system: it reads
$$\{ m\dot{r} _a , m\dot{r} _b \} = \frac{e}{c}\epsilon _{abc} B_c$$

I am, however, stumped on how this Poisson bracket has been computed. I presume r_a and A_a(r) are my canonical coordinates, and I have $$\dot{r}_a = p_a - \frac{e}{c}A_a (r)$$ with $$A_a = \frac{1}{2}\epsilon _{abc}B_br_c$$

Unfortunately, my calculations on paper aren't getting anywhere! Could someone please shed some light here? I suspect something is wrong with the canonical coordinates I'm trying to use to do the derivatives for the Poisson brackets, or maybe I'm getting my indices muddled.

Thanks :)

 

[Dickfore]

Notice that:
$$\left\lbrace p_a, f(\mathbf{r}) \right\rbrace = \sum_{b}{\left(\frac{\partial p_a}{\partial x_b} \, \frac{\partial f(\mathbf{r})}{\partial p_b} - \frac{\partial p_a}{\partial p_b} \, \frac{\partial f(\mathbf{r})}{\partial x_b}\right)} = -\frac{\partial f(\mathbf{r})}{\partial x_a}$$

Therefore:
$$\begin{align*} \left\lbrace m \, \dot{r}_a, m \, \dot{r}_b \right\rbrace & = \left\lbrace p_a - \frac{e}{c} \, A_a(\mathbf{r}), p_b - \frac{e}{c} \, A_b(\mathbf{r}) \right\rbrace \\ & = \left\lbrace p_a, p_b \right\rbrace - \frac{e}{c} \, \left\lbrace p_a, A_b(\mathbf{r})\right\rbrace + \frac{e}{c} \, \left\lbrace p_b, A_a(\mathbf{r}) \right\rbrace + \left(\frac{e}{c} \right)^2 \, \left\lbrace A_a(\mathbf{r}), A_b(\mathbf{r}) \right\rbrace \end{align*}$$

Two of these are identically zero, and for two of them you can use the hint I gave in the beginning. Then, use the properties of the Levi-Civita symbol and the definition of curl of a vector potential.

 

[joriarty]

Thanks for your help Dickfore.

I'm not sure where your first expression comes from. I suppose A_a(r) is f(r), but what is x_a? Thus, I'm having trouble seeing how to apply that to where you've broken down the Poisson brackets by linearity (I see that the first and last terms there are zero, though).

Classical mechanics usually makes sense to me, but the whole Hamiltonian formalism just isn't clicking very well...

 

[Dickfore]

$x_{a}$ stands for the a-th component of the position vector, just a $p_{a}$ is the a-th component of the momentum vector.

 

[paranormal]

I understand that Poisson brackets are a mathematical tool used in classical mechanics to describe the behavior of a system in terms of its coordinates and momenta. In this case, the system in question is a particle in a magnetic field, and the Hamiltonian is used to describe its motion.

The Poisson bracket structure for this system is computed using the canonical coordinates and momenta, which in this case are r and p, respectively. The notation \dot{r} _a refers to the derivative of the coordinate r with respect to time, and in this case, it is equal to the momentum p_a minus the vector potential A_a(r). This vector potential, in turn, is defined as A_a = \frac{1}{2}\epsilon _{abc}B_br_c, where \epsilon _{abc} is the Levi-Civita symbol and B is the magnetic field.

To compute the Poisson bracket, we use the definition \{ f,g \} = \sum_i \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}, where f and g are any two functions and q_i and p_i are the coordinates and momenta, respectively.

In this case, the Poisson bracket between the coordinates \dot{r} _a and \dot{r} _b is equal to \frac{e}{c}\epsilon _{abc} B_c, as shown in the excerpt. This result can be derived by taking the appropriate derivatives and using the definition of the vector potential.

I hope this explanation helps clarify the computation of the Poisson bracket for a particle in a magnetic field. It is important to note that Poisson brackets are a mathematical tool used in classical mechanics and may not have a direct physical interpretation. However, they are useful in analyzing and understanding the behavior of systems in terms of their coordinates and momenta.