How Can I Verify a Canonical Transformation Using the Poisson Bracket?

[skrtic]

Homework Statement

Verify that

q_bar=ln(q^-1*sin(p))

p_bar=q*cot(p)


* represents muliplication

sorry i don't know how to use the programming to make it look better



2. The attempt at a solution

my problem is that i really have no clue what is going on. I have read the section, reread the section, then looked on online just to try and find an example. I am much more of a visual learner so reading doesn't help all the time.

I guess I'm looking for some guidance of what/how to do. and not even this proble, but just an example or process.

 

[malawi_glenn]

a canonical transformation preserves the poission bracket

i.e the possion bracket of p and q: {q,p}_(p,q) = 1

thus if {q_bar, p_bar}_(p,q) = 1, then it is a canonical transformation.

(there are more ways to show it, like if there exists a generation function.. but I like the poission bracket the most, it is easy to remember)

The poission bracket is defined as
$$\left\lbrace f,g \right\rbrace _{(q,p)} = \dfrac{\partial f}{\partial q}\dfrac{\partial g}{\partial p} -\dfrac{\partial f}{\partial p}\dfrac{\partial g}{\partial q}$$