Re: Local vs Global Constants of Motion
- From: "Chris H. Fleming" <chris_h_fleming@xxxxxxxxx>
- Date: Sat, 12 Apr 2008 13:30:47 +0000 (UTC)
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.
Do I have this right?
Thanks,
Dan
--
Dan Becker
Tangent has no global inverse function as it isn't bijective. Such a
thing is impossible to construct. However you can find some local
domain where Tangent is bijective and define a local inverse function.
That is what is on your calculator.
http://en.wikipedia.org/wiki/Inverse_function#Partial_inverses
.
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- Local vs Global Constants of Motion
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