Finding the generator of a transformation

1. Jul 4, 2015

Dazed&Confused

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. Jul 4, 2015

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.