Poisson Bracket Notation: Understanding Conflicting Values - Tips & Tricks

[M. next]

In some places I see that poisson bracket of {q$_{i}$,p$_{j}$} = 1 (if i=j)
and other times I see that this same bracket equals to -1. This is very confusing! What to do?

 

[vanhees71]

It's confusing, but almost trivial. It's just a matter of definition of the Poisson bracket. I'm used to the definition

$$\{A,B\}=\frac{\partial A}{\partial q^j} \frac{\partial B}{\partial p_j} - \frac{\partial A}{\partial p_j} \frac{\partial B}{\partial q^j},$$

where tacid summation over the index $j$ is understood (Einstein convention).

If you apply this definition to your example, you get

$$\{q^j,p_k \}=\delta_{k}^{j}.$$

I could have defined the Poisson bracket as well with the opposite sign, and then I'd have gotten an additional minus sign on the right-hand side of the previous equation.

This is something one has to live with: Sometimes there are many equivalent definitions of the same mathematical structure, and when changing from one textbook to another or reading scientific papers you always have to check carefully, how the authors have chosen their convention.

This is most confusing in relativity, where it all starts with the sign of the "metric". In General Relativity you have also additional relative sign changes for the curvature, Ricci, and Einstein tensors. As I said, that's confusing, but one has to live with it unfortunately.

 

[M. next]

Wow! Okay then, you know I thought that I have a bad resource in my hands, thanks loads for making this rational.
Just one more thing: How to calculate for instance {L$_{x}$,P$_{y}$}?
Where L$_{x}$ is the angular momentum, & P$_{y}$ is the linear momentum?
I want to use the formula, but I get all confused AGAIN in the subscripts. I do not know about tensors if this has anything to do with it.
In other words, the Ps and Qs in the denominator in the formula that you're used to use, do I place instead of the subscript j an "x" or "y" or what?

ThanksBigThanks!

 

[Muphrid]

Yes, $p_k$ can stand for the x, y, or z components of momentum. We usually refer to them by numbers (i.e. $p_1, p_2, p_3$) because then you can use a summation. Ultimately, though, it still just refers to components of a vector.

 

[M. next]

Hmm, can u elaborate please? What do u mean by the "k" subscript? And then I want to know how to write it exactly? What is subscript "j"??

 

[Muphrid]

We say that a vector like $p$ can be written in component form as $(p_1, p_2, p_3)$. In cartesian coordinates $p_1$ stands for the x-component of momentum generally, but you might use spherical coordinates instead and choose for it to represent the radial component instead. Using numbered subscripts instead of letters denoting the coordinate system just makes it easier to talk about formulas without regard to the coordinate system.

We use index variables like $j$ or $k$ to stand in for any integer 1-3. That way, $p_k$ represents some component of momentum. You don't know which one because k could be 1, 2, or 3.

 

[M. next]

Oh, probably my question was not clear enough. I am not capable of asking it directly, then if you may expand this for me: {L$_{x}$, P$_{y}$}. Let us take it from here, I have an upcoming exam and I am tired of these subscripts!

 

[Muphrid]

Let's go with vanhees's definition of the bracket:

$$\{A,B\} = \frac{\partial A}{\partial q^j} \frac{\partial B}{\partial p_j} - \frac{\partial A}{\partial p_j} \frac{\partial B}{\partial q^j}$$

What this comes out to is

Let's go with vanhees's definition of the bracket:

$$\{A,B\} = \frac{\partial A}{\partial q^j} \frac{\partial B}{\partial p_j} - \frac{\partial A}{\partial p_j} \frac{\partial B}{\partial q^j}$$

What this comes out to is

$$\begin{eqnarray}<br /> \{A,B\} &=& \frac{\partial A}{\partial q^x} \frac{\partial B}{\partial p_x} - \frac{\partial A}{\partial p_x} \frac{\partial B}{\partial q^x} \\<br /> &+& \frac{\partial A}{\partial q^y} \frac{\partial B}{\partial p_y} - \frac{\partial A}{\partial p_y} \frac{\partial B}{\partial q^y} \\<br /> &+& \frac{\partial A}{\partial q^z} \frac{\partial B}{\partial p_z} - \frac{\partial A}{\partial p_z} \frac{\partial B}{\partial q^z}<br /> \end{eqnarray}$$

 

[M. next]

Ohh and what I easily do is replace each A with L$_{x}$ and B with P$_{y}$! Right? Thank you loads, you know this was very helpful. It was very nice of you!

 

[M. next]

I am gnna move the page again one more so I wouldn't have to post a new thread, Poisson's Bracket in polar coordinates! I ran through it now. If I already found let's say {H,L$_{y}$} in the normal coordinates we just talked about. They ask me again to find the Bracket (the same one) but now in r and θ?
What differs now?