Finding the generator of a transformation

[Dazed&Confused]

Homework Statement

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).

Homework Equations

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.

 

[TSny]

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.