Poisson Brackets: A Simple Example in Classical Mechanics

[snoopies622]

Could someone show me a simple example of the usefulness of Poisson brackets - for instance, a problem in classical mechanics? I know the mathematical definition of the Poisson bracket, but from there the subject quickly seems to get very abstract.

 

[snoopies622]

Just thought I'd give this a bump before it disappeared over the page one horizon for good.

(80 views and no replies!)

 

[diazona]

Well, there's always Hamilton's equation:
$$\frac{\mathrm{d}f}{\mathrm{d}t} = \bigl\{f,H\bigr\} + \frac{\partial f}{\partial t}$$
This governs the time evolution of any function f of canonical variables. It applies equally well to quantum mechanics if you replace the Poisson brackets with commutators.

If you really want a specific problem as an example, I'm sure I or someone else could look one up, but I couldn't give you one off the top of my head

 

[snoopies622]

Thanks, diazona.

Maybe what I should have asked is, what might f be? I know that it's supposed to be a function of p and q, and not energy since that's what H is. What's another function of p and q?

 

[Petr Mugver]

When you consider canonical transformations x = (p,q) -> X = (P,Q) (that preserve Hamilton's equations), you have the fundamental requirement (one-dimensional case for simplicity) [q,p]=1 -> [Q,P]=1.

A function L(p,q) (energy, angular momentum, linear momentum,...) is a constant of the motion if and only if it commutes with the hamiltonian [L,H]=0.

If A and B are constants of motion, so is [A,B].

In elementary quantum mechanic, the Poisson brakets are substituted by commutators [A,B]=AB-BA, that have the same properties of PB.

Etc.

 

[diazona]

Maybe what I should have asked is, what might f be? I know that it's supposed to be a function of p and q, and not energy since that's what H is. What's another function of p and q?

Petr gave you some good examples, like angular momentum and linear momentum (I guess that's just p)... but pretty much any physical quantity you can think of can be expressed as some function of q and p in a particular problem. Velocity, angular velocity, position, electric and magnetic fields and potentials, etc.

 

[snoopies622]

As far as usefulness goes, if I take a simple mechanical system like a pendulum, a mass on a spring or a planet orbiting a massive star, I know how to use

$$<br /> \frac {\partial L} {\partial q} - \frac {d}{dt} \frac {\partial L}{\partial \dot {q} } =0<br />$$

and

$$<br /> \dot {p} = - \frac {\partial H}{\partial q} \hspace {10 mm} \dot {q} = \frac {\partial H}{\partial p}<br />$$

to find equations of motion. But are these also situations where

$$\frac{\mathrm{d}f}{\mathrm{d}t} = \bigl\{f,H\bigr\} + \frac{\partial f}{\partial t}$$

might reveal something meaningful as well? What might f represent in these cases? Sorry for my lack of imagination - I really don't know.

(Edit: What I meant was, might the last equation reveal something that the ones above it do not?)