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[itex]\omega[/itex]x)2 , [itex]\omega[/itex]2=k/m So transforming to (Q,P) whilst setting am=-1 you get the nice equation: H = P(Q2+[itex]\omega[/itex]2) Should I now use hamiltons equation in the new coordinate basis, i.e.; dH/dP = Q2+[itex]\omega[/itex]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.