Local vs Global Constants of Motion

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

.

Relevant Pages

  • Re: Trajectory
    ... What is the trajectory of an electron circling an atom in a spaceship ... by being in a moving system and has _various_ time dilations effecting it ... it's counterpart the Ether length contraction concept. ... motion, ...
    (sci.physics.relativity)
  • Re: Trajectory
    ... What is the trajectory of an electron circling an atom in a spaceship ... by being in a moving system and has _various_ time dilations effecting it ... it's counterpart the Ether length contraction concept. ... motion, ...
    (sci.physics.relativity)
  • Re: angle n speed of golf ball
    ... As ImageAnalyst points out, ... the apparent motion of the ball wrt time. ... expressing those displacements in the equations for motion ... and solving for the intial speed and trajectory angle at ...
    (comp.soft-sys.matlab)
  • Re: Local vs Global Constants of Motion
    ... constants of motion. ... Clearly you could do this for any *particular* trajectory. ... Tangent has no global inverse function as it isn't bijective. ... domain where Tangent is bijective and define a local inverse function. ...
    (sci.physics.research)