Re: [Phys-l] question about coupled oscillators

[John Denker <jsd@av8n.com>]

On 11/03/2006 06:14 PM, Carl Mungan wrote:
> In writing down the assumed form for the displacement of each 
> oscillator i in a coupled system, in general one includes a phase 
> constant d_i in the argument of the cosine:
>
> x_i (t) = A_i cos (w*t + d_i)
>
> However, in all the examples I can think of, the relative phases of 
> the oscillators *in a single normal mode* always turn out to be 
> either 0 or pi. Why can't we get other values?

Wow, that's a thought-provoking question.




 ** Spoiler Alert! **
 ** Stop reading now, if you want to figure it out on your own. **









Here's how I thought about it.  This is pretty much a "stream of
consciousness" account.

1) Any phase other than 0 or pi will break a certain type of
simple time-reversal symmetry.

2a) Usually when we think of normal modes, we think of standing waves,
which will usually (!) be time reversible.

2b) But are there exceptions?  You could have an exception if you
had a normal mode that involves some sort of /chiral/ motion,
perhaps the normal modes of a DNA molecule or Wilberforce pendulum.

3) Ah, but there's a simpler way to cobble up an exception.
Consider the normal modes of a tub of water.
  http://www.av8n.com/physics/orbitals.htm

In particular, consider the 2px mode.  It is a nice standing wave.
The 2py mode is also a nice standing wave.  We haven't done anything
exciting so far.

It gets exciting when we form the 2p+ mode, where 2p+ ~ 2px + i 2py.
That is a /running/ wave.  It runs around and around the tub, counter
clockwise.  This 2p+ mode is perfectly usable as a "normal" mode.

  Note we are assuming the original two orbitals (2px and 2py) are
  degenerate.  That means we can form a whole family of orbitals
  all with the same energy.  The family includes 2px, 2py, 2p+, 2p-,
  and others.

When the 2p+ mode is active, you can get any relative phase you want,
just by selecting appropriate points and comparing their phase.
Delta azimuth = delta phase.