Configuration Spaces

[theexi...@gmail.com]

Section 1.2
Link: http://mitpress.mit.edu/SICM/book-Z-H-9.html#%_sec_1.2

I have a few questions about these.... I'll bold the parts of two
exepts that confused me.

"For example, a system consisting of two point particles constrained to
move in three dimensions so that the distance between the particles
remains fixed has a five-dimensional configuration space: thus with
three numbers we can fix the position of one particle, <b>and with two
others we can give the position of the other particle relative to the
first.</b>"

&

"The dimension of the configuration space of the juggling pin is six:
<b>the minimum number of parameters that specify the position in space
is three, and the minimum number of parameters that specify an
orientation is also three.</b>"

Do those statements contradict each other? In the first exerpt, it says
the configuration space is 5 dimensions yet in the second one it says
the minimum number is 6...

Another question: what would be the two other parameters in the first
exerpt? I was thinking it would just be two angles because the distance
isn't needed because it is fixed. In the other exerpt, what are the 3
parameters that specify orientation and why is the minimum number 3?

Thanks for all the help!

[Tobin Fricke]

On Wed, 17 Aug 2005, theexi...@gmail.com wrote:

> "For example, a system consisting of two point particles constrained to
> move in three dimensions so that the distance between the particles
> remains fixed has a five-dimensional configuration space: thus with
> three numbers we can fix the position of one particle, <b>and with two
> others we can give the position of the other particle relative to the
> first.</b>"

The dimensionality of the configuration space is the number of numbers
necessary (and sufficient) to specify the state of the system. If you
have one point particle in three dimensions, you need three numbers (i.e.,
{x, y, z} or {r, phi, theta}, etc). Two point particles, each free to be
anywhere, needs six numbers. For instance, you could give {x,y,z}
coordinates for each particle.

If we have the requirement that the particles have a fixed distance
between them, then we can no longer freely specify all six coordinates,
because there is some relationship between them. As you say below, you
can specify the relative position of the 2nd particle with only two
coordinates -- so you have a total of five instead of six for the two
particles with a fixed relative distance.

> "The dimension of the configuration space of the juggling pin is six:
> <b>the minimum number of parameters that specify the position in space
> is three, and the minimum number of parameters that specify an
> orientation is also three.</b>"

A juggling pin is different from a point particle in that it has an
orientation. In order to specify the configuration of a bowling pin, you
have to specify how it is oriented in space (point up, sideways, etc). A
point particle, however, is spherically symmetric--all orientations are
identical and indistinguishable, so there is no point in specifying the
orientation (indeed, you can't).

In general the orientation of an asymmetric object in three-dimensional
space is specified with three numbers (one way is with Euler Angles,
http://en.wikipedia.org/wiki/Euler_angles). If the bowling pin is
cylindrically symmetric (aren't they?) then you can specify the
orientation with 2 numbers; if not, it takes three numbers. Add the
numbers which give the position and you get a total of either 5 or 6
numbers.

> Do those statements contradict each other? In the first exerpt, it says
> the configuration space is 5 dimensions yet in the second one it says
> the minimum number is 6...

The configuration space depends on the system you are configuring... they
are different systems, so they have different configuration spaces.

> Another question: what would be the two other parameters in the first
> exerpt? I was thinking it would just be two angles because the distance
> isn't needed because it is fixed. In the other exerpt, what are the 3
> parameters that specify orientation and why is the minimum number 3?

The angles would work. In the second case, you can give the orientation
with Euler angles. The important thing is that there are many different
ways of specifying the configuration, but it's always with the same number
of numbers (coordinates). For instance, whether we specify the position
of a point with cartesian coordinates {x,y,z} or with spherical
coordinates {rho, theta, phi}, we still need three numbers. And by
changing the origin of the coordinate system we get infinitely more
possible coordinates systems.

Tobin

[jao]

Hi, just joined and, although i am not walking the book sequentially, i
thought it'd be fun reading about it :-)

Regarding your questions,

> Do those statements contradict each other? In the first exerpt, it says
> the configuration space is 5 dimensions yet in the second one it says
> the minimum number is 6...

No, by no means: we're talking about two _different_ dynamic systems:
two point particles in the first case, a rigid solid in the second.
Note that if you try to model the juggling pin as 2 point particles
(say, it's end points), you're missing one dimension: the pin can
rotate along the axis joinning its end-points; that's where the extra
dimension comes from.

> Another question: what would be the two other parameters in the first
> exerpt? I was thinking it would just be two angles because the distance
> isn't needed because it is fixed.

Yup, thats right. First, you need three coordinates to locate the first
particle. Then, the second one must be at a fixed distance, that is,
anywhere on a sphere centered at the first one. That's like the surface
of the earth: you give longitude and latitude to fix a point on it.

> In the other exerpt, what are the 3
> parameters that specify orientation and why is the minimum number 3?

These are usually known as Euler angles (well, in fact, Euler's are
just one among an infinite number of possible orientation coordinates).
You first give the longitude, then the latitude and finally you must
give an angle of rotation across the axis (this happens because you've
got an extended solid here). You'll find a much better explanation
here:

http://en.wikipedia.org/wiki/Euler_angles

Hope this helps,
jao

[Tobin Fricke]

On Wed, 17 Aug 2005, jao wrote:

> Note that if you try to model the juggling pin as 2 point particles
> (say, it's end points), you're missing one dimension: the pin can
> rotate along the axis joinning its end-points; that's where the extra
> dimension comes from.

I think the difficulty might be in whether we consider this an "ideal"
bowling pin (cylindrically symmetric) or a "real" one (not quite). (-:

Welcome.

Tobin

[jao]

> I think the difficulty might be in whether we consider this an "ideal"
> bowling pin (cylindrically symmetric) or a "real" one (not quite). (-:

Hmm, not sure if i understand what you mean, but just in case:
symmetries do not affect the number of degrees of freedom. Even an
ideal cylindrically symmetric body has six degrees of freedom, not
five. You cannot ignore its ability to rotate along its symmetry axis
when analysing its movement. Just think of its moment of inertia along
that axis, or the kinetic energy it will have while rotating along the
same axis: you need a coordinate describing a rotation along the axis
to specify its angular velocity!.

When you disregard this extra degree of freedom, you don't do it
because of symmetry (you can't), but because it actually doesn't exist
due to its being infinitely thin (which is the case of the two
particles at fixed distance).

In other words, symmetries do not affect the dimensionality of the
configuration space. But, of course, symmetries will allow simplifying
the equations of motion in that they will be independent of some
(properly chosen) coordinates.

Sorry if all this was already matter-of-fact :)

jao

[theexi...@gmail.com]

Thanks for the help :) I didn't catch the part where it said it was two
point coordinates... I appreciate the explainations, I understand it now