Re: [Phys-l] question about coupled oscillators

[Jack Uretsky <jlu@hep.anl.gov>]

Hi all-
	The coupled oscillator problem is discussed at length in Olenick, 
et al. <The Mechanical Universe> (NOT the advanced edition), Ch 22.
	The problem of 2 coupled oscillators can be worked out easily, 
TKarim's example is OK (though the conclusion is erroneous).  The paradigm 
equation is (' stands for time derivative, f^2 means f squared)
	x'' + f1^2 x +k(x-y) =0
	y'' + f2^2 y +k(y-x) =0

	The eigenvalue equation is a quadratic in the square of the 
eigenfrequencies; f1, f2, k are real constants. It is easy to show that
both squared eigenfrequencies are real and positive.
	The eigenvectors are two dimensional column vectors, one for each 
(squared) eigenfrequency.  The eigenvalue condition gives 1 relation 
between the two components of each eigenvector.  The normalization 
condition requires the squared magnitudes of the two components to sum to 
unity.  Since only the magnitudes are involved, there is an undetermined, 
arbitrary phase for each eigenvector.  The relative phases may be 
determined from the starting conditions on x and y.  Different phases may 
refer to different starting conditions, but 0 and/or pi are certainly not 
mandated in general.
				Regards,
					Jack


On Wed, 8 Nov 2006, Karim Diff wrote:

> 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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