Re: Local vs Global Constants of Motion

On Apr 8, 2:22 pm, d...@xxxxxxxxxxx wrote:
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.

Physically, the phase doesn't change when you add 2pi to it. Therefore,
we must identify phi with phi+2pi. Topologically, the phase variable
lives on a circle. For example, its the same circle as the result of
identifying the endpoints of the [0,2pi] interval. The identify function
on this interval, basically the phase variable phi, is not continuous.
It cannot be continuous for the same reason that the circle needs an
atlas of at least two charts to fully parametrize it.

Note, however that cos(phi) and sin(phi) are perfectly continuous,
smooth functions of the physical phase. They are as well globally
defined as the energy.

Hope this helps.

Igor

.

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