Canonical transformation for Harmonic oscillator

[aaaa202]

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are:
Q = ap/x , P=bx²
And apply the transformation to the harmonic oscillator.
I did the first part and found a = -1/2b
I am unsure about the next part tho:
We have the hamiltonian in (x,p):
H = 1/2m(p²+(m$\omega$x)² , $\omega$²=k/m
So transforming to (Q,P) whilst setting am=-1 you get the nice equation:
H = P(Q²+$\omega$²)
Should I now use hamiltons equation in the new coordinate basis, i.e.;
dH/dP = Q²+$\omega$² = dQ/dt
dH/dQ = 2QP = -dP/dt
And solve these differential equations for Q,P and transform back? I think so but these equations are just not very easu. I mean the second one is easy to do by means of substitution but really the first one is just a mess. What is the smartest way to do this? I can't really get the first one solved.

 

[gabbagabbahey]

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are:
Q = ap/x , P=bx²
And apply the transformation to the harmonic oscillator.
I did the first part and found a = -1/2b

Do you mean $a=-\frac{1}{2b}$, or $a=-\frac{1}{2}b$? Brackets are important!

I am unsure about the next part tho:
We have the hamiltonian in (x,p):
H = 1/2m(p²+(m$\omega$x)² , $\omega$²=k/m

Again, brackets are needed if you meant $H=\frac{1}{2m}\left(p^2+(m\omega x)^2\right)$

So transforming to (Q,P) whilst setting am=-1 you get the nice equation:
H = P(Q²+$\omega$²)

(1) Why are you setting am=-1?
(2) You don't get that when you set am=-1

Without setting am=-1, you should get $H=-\frac{1}{4m}PQ^2 -am\omega^2P$

Should I now use hamiltons equation in the new coordinate basis, i.e.;
dH/dP = Q²+$\omega$² = dQ/dt
dH/dQ = 2QP = -dP/dt
And solve these differential equations for Q,P and transform back? I think so but these equations are just not very easu. I mean the second one is easy to do by means of substitution but really the first one is just a mess. What is the smartest way to do this? I can't really get the first one solved.

The second DE is only easy to solve once you know Q. The first one is seperable, so al that is needed is some striaghtforward integration... try a trig substitution if you are stuck on the integration part