Re: [Phys-l] question about coupled oscillators

[Carl Mungan <mungan@usna.edu>]

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


--
Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/