Re: [Phys-l] question about coupled oscillators

[Karim Diff <karim.diff@sfcc.edu>]

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
> > > > >           
>
>
>