1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hamiltonian, generating function, canonical transformation

  1. Mar 4, 2012 #1

    fluidistic

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m.
    Let the transformation (p,q) -> (Q,P) be such that [itex]F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta[/itex].
    1)Find [itex]K(Q,P)[/itex] where [itex]\theta[/itex] is a function of time and K is the Hamiltonian in function of Q and P.
    2)For which values of [itex]\theta (t)[/itex] does [itex]K(Q,P)[/itex] vanishes?
    2. Relevant equations
    [itex]H=\frac{p^2}{2m}+\frac{\omega m }{2}[/itex].
    [itex]p=\frac{\partial F_2}{\partial q}, Q=\frac{\partial F_2}{\partial P}[/itex]
    [itex]K=H+\frac{\partial F_2}{\partial t}[/itex]
    3. The attempt at a solution
    I guess the main idea is to calculate [itex]\frac{\partial F_2}{\partial t}[/itex] and express p and q in terms of P and Q.
    Playing with the relevant equations I get that [itex]q=Q\cos \theta +m \omega P \sin \theta[/itex], [itex]p=\frac{P}{\cos \theta }-m\omega \tan \theta (Q\cos \theta + m\omega P \sin \theta )[/itex].
    Also, [itex]\frac{\partial F_2}{\partial t}=qP\dot \theta \sin \theta - \frac{m\omega }{2}(q^2+P^2)\left ( \frac{\dot \theta }{\sin ^2 \theta } \right ) [/itex].
    This gave me [itex]K=\frac{1}{2m} \left [ \frac{P}{\cos \theta }-m\omega (Q\cos \theta+m \omega \sin \theta ) \tan \theta \right ] ^2+\frac{m\omega}{2}(Q\cos \theta + m \omega P \sin \theta )^2+ (Q \cos \theta + m \omega P \sin \theta )P\dot \theta \sin \theta -\frac{m\omega }{2} [(Q\cos \theta + m\omega P \sin \theta)^2+P^2]\left ( \frac{\dot \theta }{\sin ^2 \theta } \right )[/itex].
    The new Hamiltonian does indeed depends on the variables it should, but it looks so horrible that I cannot believe I made things right. What am I missing?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?