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Configuration Space In Classical Mechanics: Definition

  1. Nov 8, 2011 #1
    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,..,q3n-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" [Broken]


    There they say

    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.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 8, 2011 #2

    Bill_K

    User Avatar
    Science Advisor

    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.
     
  4. Nov 10, 2011 #3
    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, thanx :)


    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 dont 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.
    tconfig%25space.JPG
     
    Last edited: Nov 10, 2011
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