Configuration Space In Classical Mechanics: Definition

[Dr_Pill]

Hi,

I'm a bit confused wit the concept Configuration Space.

First, the professor defined generalised coordinates as such:

U got a system of n particles, each particle has 3 coordinates(x,y,z), so u got 3n degrees of freedom.
If the system has k holonomic constraints, u got 3n-k degrees of freedom.
Instead of working with cartesian coordinates, we now define a new set of coordinates q₁,q₂,..,q_3n-k.

These are the generalised coordinates of the system,3n-k in total.

I get this.

Then a little bit further, when explaining Hamilton's Variatonal Principle, he defines a Configuration Space.

"The configuration space of a system is a 3n-k dimensional space with the generalised coordinates on the coordinate-axes."

So far, so good.

On the reference list of this course,Classical Mechanics of Goldstein is listed.

First page of the second chapter of Goldstein:

This n-dimensional space is therefore known as the configuration space...

In classical mechanics from Kibble, I didn't even found such thing as config space.

Also, on the internet I've found another course of Classical Mechanics:

http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH2A.pdf"


There they say

Meaning of “motion of system between time t1 and t2”:
• A system is characterized by n generalized coordinates
q1,q2,q3,..qn.
• At time t1: q1(t1),q2(t1),..,qn(t1) represent a point in the ndimensional
configuration space.
• As time goes on, the system point moves in configuration
space tracing out a curve, called the
path of motion of the system.
• At time t2: q1(t2),q2(t2),.. ,qn(t2)
represent another point in the ndimensional
configuration space.

Here they say n generalised coordinates in n dimensional space, not like according to my professor 3n-k dimensions with 3n-k generalised coordinates!
Also, there's a little graph with on the horizontal axis q1 and on the vertical axis q2, but there are n dimension, according to their course !
But for the axes only q1 and q2 is used, so why not qn-1 and qn.
But a graph with only two axis, is 2-dimensional right?
It is not ndimensional

See my frustration here?

Please help me.

 

[Bill_K]

The dimensionality of configuration space is always equal to the number of degrees of freedom. The references you've cited are just using the symbol n to denote different things. Your professor is describing N particles moving in three dimensions with k holonomic constraints, so the number of degrees of freedom is n = 3N - k.

 

[Dr_Pill]

I thought so myself.My professor made some mistakes, he used little n instead big N for the particles, very confusing at first.

Now is it clear, thanks :)


Edit:

Still not 100% clear:

Given is a simple graph of the configuration space with the generalised coordinates q1 and q2 on the axes, this is only 2-dimensional right, that I don't get?

Why put only those generalised coordinates on the axes? Is it equivalent to a simple x and y axes?

I need this to define the path of motion of the system.

U define a system that has N particles, so N generalised coordinates.But system has 3N-K dimensions, so u need 3N-k axes, and this is impossible to plot?

This is the simple graph below, I found it on the net.It is from Texas University.