# Poisson Brackets

1. Nov 13, 2009

### Nusc

1. The problem statement, all variables and given/known data

$$f(p(t),q(t)) = f_o + \frac{t^1}{1!}\{H,f_o\}+\frac{t^2}{2!}\{H,\{H,f_o\}}+...$$
Prove the above equality. p & q are just coords and momenta

How do we do this if we don't know what H is?
Where do we start?
2. Relevant equations

3. The attempt at a solution

2. Nov 13, 2009

### jdwood983

Looks like you may not need to know what the Hamiltonian is, just how the Hamiltonian works in the Poisson bracket (assume $f$ is not a constant of motion):

$$\frac{df(q,p)}{dt}&=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial q}\frac{dq}{dt}+\frac{\partial f}{\partial p}\frac{dp}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial H}{\partial q}=\frac{\partial f}{\partial t}+\{f,H\}$$

So, using this, it looks like you would expand $f(q,p)$ as a Taylor series. Note that if $f(q,p)$ is a constant of motion, $f_{,t}=0$ so that you have

$$\frac{df(q,p)}{dt}=\{f,H\}$$

3. Nov 13, 2009

### Nusc

What's the theorem that says mixed partials are commutative? Not Claurait's....

4. Nov 13, 2009

### Nusc

Also

I know

$$\frac{\partial }{\partial t}\{H,f\} = \{\frac{\partial H}{\partial t},f\} + \{H,\frac{\partial f}{\partial t}\}$$

$$\frac{d}{d t}\{H,f\} =?$$

5. Nov 14, 2009

No, if $f(q,p)$ is a constant of motion, then $$\frac{df}{dt}=0[/itex]. The fact that $f$ has no explicit time dependence (it is given as a function of $q$ and $p$ only), tells you that [tex]\frac{\partial f}{\partial t}=0[/itex] 6. Nov 14, 2009 ### gabbagabbahey You tell us...expand the Poisson bracket and calculate the derivatives...what do you get? 7. Nov 14, 2009 ### Nusc [tex] \frac{d}{dt}\frac{\partial f}{\partial q}\frac{\partial H}{\partial p}-\frac{d}{dt}\frac{\partial f}{\partial p}\frac{\partial H}{\partial q}$$

8. Nov 16, 2009

### gabbagabbahey

Okay, now use the product rule...

[tex]\frac{d}{dt}\left(\frac{\partial f}{\partial q}\frac{\partial H}{\partial p}\right)=[/itex]

???