1.2  Configuration Spaces

Let us consider mechanical systems that can be thought of as composed of constituent point particles, with mass and position, but with no internal structure.17 Extended bodies may be thought of as composed of a large number of these constituent particles with specific spatial relationships among them. Extended bodies maintain their shape because of spatial constraints among the constituent particles. Specifying the position of all the constituent particles of a system specifies the configuration of the system. The existence of constraints among parts of the system, such as those that determine the shape of an extended body, means that the constituent particles cannot assume all possible positions. The set of all configurations of the system that can be assumed is called the configuration space of the system. The dimension of the configuration space is the smallest number of parameters that have to be given to completely specify a configuration. The dimension of the configuration space is also called the number of degrees of freedom of the system.18

For a single unconstrained particle it takes three parameters to specify the configuration. A point particle has a three-dimensional configuration space. If we are dealing with a system with more than one point particle, the configuration space is more complicated. If there are k separate particles we need 3k parameters to describe the possible configurations. If there are constraints among the parts of a system the configuration is restricted to a lower-dimensional space. 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, and with two others we can give the position of the other particle relative to the first.

Consider a juggling pin. The configuration of the pin is specified if we give the positions of the atoms making up the pin. However, there exist more economical descriptions of the configuration. In the idealization that the juggling pin is truly rigid, the distances among all the atoms of the pin remain constant. So we can specify the configuration of the pin by giving the position of a single atom and the orientation of the pin. Using the constraints, the positions of all the other constituents of the pin can be determined from this information. The dimension of the configuration space of the juggling pin is six: 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.

As a system evolves with time, the constituent particles move subject to the constraints. The motion of each constituent particle is specified by describing the changing configuration. Thus, the motion of the system may be described as evolving along a path in configuration space. The configuration path may be specified by a function, the configuration-path function, which gives the configuration of the system at any time.

Exercise 1.2.  Degrees of freedom
For each of the mechanical systems described below, give the number of degrees of freedom of the configuration space.

a.  Three juggling pins.

b.  A spherical pendulum, consisting of a point mass hanging from a rigid massless rod attached to a fixed support point. The pendulum bob may move in any direction subject to the constraint imposed by the rigid rod. The point mass is subject to the uniform force of gravity.

c.  A spherical double pendulum, consisting of one point mass hanging from a rigid massless rod attached to a second point mass hanging from a second massless rod attached to a fixed support point. The point mass is subject to the uniform force of gravity.

d.  A point mass sliding without friction on a rigid curved wire.

e.  A top consisting of a rigid axisymmetric body with one point on the symmetry axis of the body attached to a fixed support, subject to a uniform gravitational force.

f.  The same as e, but not axisymmetric.

17 We often refer to a point particle with mass but no internal structure as a point mass.

18 Strictly speaking, the dimension of the configuration space and the number of degrees of freedom are not the same. The number of degrees of freedom is the dimension of the space of configurations that are ``locally accessible.'' For systems with integrable constraints the two are the same. For systems with non-integrable constraints the configuration dimension can be larger than the number of degrees of freedom. For further explanation see the discussion of systems with non-integrable constraints in section 1.10.3. Apart from that discussion, all of the systems we consider have integrable constraints (they are ``holonomic''). This is why we have chosen to blur the distinction between the number of degrees of freedom and the dimension of the configuration space.