Natural Science Forum / Physics / Research / April 2008

I'm trying to understand the distinction between local and global
constants of motion. Thirring's Classical Mathematical Physics (http://
books.google.com/books?id=NFdStDHPOfcC pg 44) gives the 1-D harmonic
oscillator as an example.

The phase-space trajectories are (x(t), p(t)) = (A sin(w t + phi), B w
cos(w t + phi)), and the constants of motion are energy = x^2 + (p/
w)^2 and phase = arctan(w q / p) - w t.

The energy is clearly global, but he says that the phase angle is only
defined locally. I see that the arctan is discontinuous across the p
axis
so that this particular expression doesn't work. But I keep thinking
that
there ought to be a way to fix this by adding some other "correcting"
discontinuous function to the expression.

Clearly you could do this for any *particular* trajectory. Say phi =
0. Then we could have something like

arctan(w q / p) - w t + ...
  {     0 for     0 < w t <= pi/2
      -pi for  pi/2 < w t <= 3pi/2
    -2 pi for 3pi/2 < w y <= 5pi/2
    etc
  }

But I think that you would need different "correcting" functions for
different initial phase angles. So you couldn't have a function of
*only*
(q,p,t) that is constant for *all* trajectories. And by a *global*
constant
of motion we mean a constant of motion that applies not only at all
times
for given trajectory, but that applies to *all* trajectories.

Do I have this right?

Thanks,

Dan
--
Dan Becker