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A general problem in (q,p) -> (Q,P) for Hamiltonian

  1. Jan 19, 2013 #1
    Hello everyone, this is my first thread. Hope to be helpful here, as well as to find some help! :D


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

    Given a Hamiltonian [itex]H(q,p)[/itex](known) and given a transformation of coordinates [itex](q,p)\rightarrow (Q,P)[/itex]:

    a) Show that it is a canonic transformation
    b) Solve the equations of motion
    c) Is this a simmetry of H? Find the constants of motion.

    2. Relevant equations

    (See attempted solution)

    3. The attempt at a solution

    Just need to know if I'm wrong in some of these, and in that case being sure I'm doing right

    a) By using the Poisson brackets, they must verify all of these

    [itex]\left\{Q,P\right\}=1[/itex]
    [itex]\left\{ P,Q \right\}=-1[/itex]
    [itex]\left\{ Q,Q \right\}=0[/itex]
    [itex]\left\{ P,P \right\}=0[/itex]


    b) I tried by using the chain rule in the Hamiltonian

    [itex]\frac{\partial H}{\partial p}=\frac{\partial H}{\partial P}\frac{\partial P}{\partial p} + \frac{\partial H}{\partial Q}\frac{\partial Q}{\partial p}[/itex]

    [itex]\frac{\partial H}{\partial q}=\frac{\partial H}{\partial P}\frac{\partial P}{\partial q} + \frac{\partial H}{\partial Q}\frac{\partial Q}{\partial q}[/itex]

    then I have a system of two equations and the two [itex]\frac{\partial H}{\partial Q}[/itex] and [itex]\frac{\partial H}{\partial P}[/itex], which will give me the equation of motion by using Hamilton's:

    [itex]-\frac{\partial H}{\partial Q}= \dot{P}[/itex]
    [itex]-\frac{\partial H}{\partial P}= \dot{Q}[/itex]


    c) This point I haven't got clear. I've tried to relate it with 1, knowing that a quantity f is a constant of motion if the Poisson Bracket [itex]\left\{ H,f \right\}=0[/itex], i.e., if commutes with the Hamiltonian, but yet I've not it clear.

    Thank you very much!
     
    Last edited: Jan 19, 2013
  2. jcsd
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