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Finding the generator of a transformation

  1. Jul 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider ##\mathscr{H} = \frac12 p^2 + \frac12 x^2, ## which is invariant under infinitesimal rotations in phase space ( the ##x-p## plane). Find the generator of this transformation (after verifying that it is canonical).

    2. Relevant equations


    3. The attempt at a solution

    So the transformation to ##\bar{x}, \ \bar{p}## is
    $$\bar{p} = p - \epsilon x, \ \bar{x} = p \epsilon + x$$.
    This is canonical as clearly ##\{ \bar{x}, \bar{x} \} = \{ \bar{p}, \bar{p} \} = 0## and $$\{\bar{x},\bar{p} \} = 1 \times 1 +\epsilon \epsilon = 1$$
    if we work only to first order in ##\epsilon##.

    The differential equations for the generator ##g## are
    $$ \frac{\partial g}{\partial x} = x, \ \frac{\partial g}{\partial p} = p.$$

    Thus ##g(x,p) = \frac12 x^2 + \frac12 p^2 + C.##

    This seems to make sense, but I'm not sure if this is correct.
     
  2. jcsd
  3. Jul 4, 2015 #2

    TSny

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    Homework Helper
    Gold Member

    I think it's correct. So, the generating function is essentially the Hamiltonian in this case (no need to keep ##C##). That should make sense if you think about the phase space trajectories for this system.
     
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