Clarification on Degrees of Freedom  (October 2, 2009)
-----------------------------------

I received the following question(s) from one of the members
of the PHY293 class. Hope my answer(s) will clarify things
for other people in class who may have been confused by the
same things.

 
> Dear Professor,
>
> In our lecture on Monday you mentioned that coupled oscillators have
> "more degrees of freedom" than the ones we had studied earlier, implying
> that there is more than one natural and resonance frequency.
>
Well, with more objects (two pendulum bobs instead of one) we have
more degrees of freedom, precisely how many I'll try to address below.

> I am having trouble understanding what you mean by "more degrees of
> freedom" and how that implies that we have two natural and resonance
> frequencies. Perhaps the trouble is in my understanding of the
> definition of degrees of freedom.
>
> The way I understood it, an ant that is forced to walk along a single
> line had only one degree of freedom. We could represent its motion by a
> single parameter. On the other hand, an ant that was free to walk only
> on a plane now can't have its motion described by a single variable, we
> would now need two. And so on for an ant that is allowed to walk in a
> volume, etc.
>
This is all true, if all the ant can do is walk at a constant speed.
When accelerations are involved (as they are in our oscillators) then
each dimension actually has two degrees of freedom -- a position and
a speed, both of which can be changing. So our 1D SHO actually already
has two degrees of freedom -- the way I'm using the term. I apologise
if this contradicts something you learned elsewhere, but I suspect
it is just building on the concept of your ant (our massive object
on a spring) to add the idea that they any can take up a position
but also a speed in 2n-dimensional phase space.


> So coming back to our coupled oscillations, are we not able to describe
> all the motion in terms of a single parameter, that we have chosen to be
> time? That's where my confusion lays, and thus can't see why we have a
> multiple degree of freedom system, let alone its implications. I would
> very much appreciate your help in this matter.
>
This is probably the confusion that I should work hardest to clarify. It
is not the natural frequency that is associated with any degree of
freedom. The frequency(ies) is(are) fixed by the equation(s) of motion.
The freedom comes in specifying the initial conditions. Each oscillating
object has two dof, usually specified by x(0) and v(0). When we go to
multiple oscillators we get multiple frequencies (eigenvalues) but we
also get 2N degrees of freedom -- ie. a multiplication of initial
conditions are needed to fully specify the system. I didn't explain
this clearly enough in class.

Prof. Trischuk