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Re: [Phys-l] question about coupled oscillators

At 11:44 AM 11/8/2006, Chuck, you 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.

[Jack]
>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
......
--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)

It's all very well for Jack to mention the eigenvalue of the NxN Jacobi matrices
- equivalent to estimating the largest zero of a sequence of orthogonal polynomials.
One could, I suppose, go further in systematizing properties of matrix model
partition functions which I define here as solutions to the Virasoro-like sets of
linear differential equations and proceed to consideration of non-Gaussian
phases of the Hermitean one-matrix model.

The underlying concept, dimly limned, is about connected correlators
like phase-independent "check-operators" acting on the small space
of T-variables which parameterize the polynomial W(z). These multidensity
check-operators with the appropriate reservations, look very similar
to the Gaussian case. This suggestion is speculative however, and may
even be a figment of my imagination.

Here is a URL, not for a physical system of chaotic phase to which
Jack refers and for which you seek examples, but rather the usual
pi phase coupling to which I previously alluded.
<http://facstaff.morehouse.edu/~cmoore/CoupledOscillations.htm>



Brian Whatcott Altus OK Eureka!