Configuration Space

an n -dimensional space, with the number of dimensions equal to the number n of degrees of freedom of a system. It is used to arbitrarily represent the motion of the entire system as the motion of some point in this space.

When a mechanical system moves with respect to some frame of reference, its configuration, that is, the position of the system itself and the relative positions of its parts, can be determined at any moment of time by the generalized coordinates q₁, q₂. … , q_n. If these coordinates are considered as n Cartesian coordinates in n -dimensional space, a definite point in this space, called the image point, will correspond to each configuration of the system. Such a space is called a configuration space. For example, in systems with one, two, or three degrees of freedom (for example, a plane mathematical pendulum, a spherical pendulum, or a free material point) the configuration space will be a line, plane, or three-dimensional space, respectively. For a free rigid body with six degrees of freedom, the configuration space will be six-dimensional.

The configuration of a moving system will continuously vary and the image point will also continuously vary its position in the configuration space, tracing a curve called by convention the trajectory of the system. Consequently, the motion of a system can be represented as the motion of the image point in the configuration space. Such a representation is used in the consideration of certain properties of a moving system, particularly properties established by a number of variational principles of mechanics.