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?
In particular, the "weak coupling" problem (where you pull just one
oscillator aside and then get beats) looks like an example of pi/2
relative phase difference between the two oscillators. Except I don't
have a single normal mode; I have two normal modes equally excited.
But there's a part of me that wants to call this something like
another normal mode since each particle is oscillating in a nicely
repetitive manner. (Okay, maybe I need to require the two normal mode
frequencies to be commensurate for this statement to be exactly
correct.) That is, I'm wanting to define "normal mode" as any
smoothly repetitive pattern, but apparently that's too loose a
definition to be right.