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I did not work it out in detail yet but what about this:
2 unequal masses & 3 horizontal springs (different k's) between 2 walls.
In crude ASCII
Wall |/\/\/\/\[ m 1 ]/\/\/\[m 2 ]/\/\/\/| Wall
The phases are 0 & pi if the two masses and the springs are the same
(basic coupled oscillators)
but I believe the phase relationship becomes more complicated in the
general case (2 different masses and 3 different k's).
Karim Diff
Carl Mungan wrote:
So if the eigenvectors can be complex even when one component is
chosen to be real, then that implies that the oscillators in a normal
mode need NOT pass through the equilibrium position at the same
instant (ie. relative phases other than 0 or pi are possible).
In that case, can someone rig up a *simple soluble* example of such
an eigenvalue problem?
To make my request clear for those who have forgotten the beginning
of this thread. I want an example of a system of coupled oscillators,
as simple as possible, where the relative phases between oscillators
in a normal mode are neither 0 nor pi. All the usual textbook
examples I can think only have in-phase or exactly-out-of-phase
relative motions of the oscillators.
The correct statement, that you seem to be striving for, is that the
eigenvalues of a Hermitean matrix (includes real, symmetric
matrices) are real. This says nothing about the scaling of the
eigenvectors, which are usually scaled by normalizing them to a
(squared) norm of unity - which leaves them with an undetermined
phase. That phase is just the subject of the present discussion.
Regards,
Jack
Is that true even if I insist that one component arbitrarily have the
real value 1, to remove the indeterminacy in the overall scaling of
an eigenvector? This way d_i is the relative phase, which is what I
meant.
The argument is erroneous. The reality condition is on the
eigenvalues, not the eigenvectors.
Jack
Having thought about it a little more, this is what I come up with:
Newton's third law says that oscillators i and j exert equal
magnitude forces on each other. This in turn will lead to elements
K_ij and K_ji in the force-constant matrix K to be equal. In turn,
this means K is symmetric (and of course it's real); and so is the
mass matrix M. But this means the eigenvectors must be real, even if
I write the oscillator displacement vector x in complex form where
the i-th component is A_i exp (i d_i). But this means d_i can only be
0 or pi.
What do you think of this argument? -Carl
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