Notes for AKT-140115/0:30:33

Configuration space Given a topological space [math]\displaystyle{ X }[/math], the [math]\displaystyle{ n }[/math]th ordered configuration space of [math]\displaystyle{ X }[/math] denoted by [math]\displaystyle{ \mathrm{Conf}_n(X) }[/math] is the set of [math]\displaystyle{ n }[/math]-tuples of pairwise distinct points in [math]\displaystyle{ X }[/math], that is [math]\displaystyle{ \mathrm{Conf}_n(X):= \prod^n X \setminus \{(x_1, \ldots, x_n) : x_i = x_j \;\mathrm{for} \;i\ne j\} }[/math].

In physics, parameters are used to define the configuration of a system and the vector space defined by these parameters is the configuration space of the system. It is used to describe the state of a whole system as a single point in a higher-dimensional space.

Examples of Configuration space

1. The configuration space of a particle in [math]\displaystyle{ \mathbb{R}^3 }[/math] is [math]\displaystyle{ \mathbb{R}^3 }[/math]. For [math]\displaystyle{ n }[/math] particles in [math]\displaystyle{ \mathbb{R}^3 }[/math], it is [math]\displaystyle{ \mathbb{R}^{3n} }[/math]

2. For a rigid body in [math]\displaystyle{ \mathbb{R}^3 }[/math], the configuration space is [math]\displaystyle{ \mathbb{R}^3 \times SO(3) }[/math]. Generally, it is [math]\displaystyle{ \mathbb{R}^n \times SO(n) }[/math], where [math]\displaystyle{ SO(n) }[/math] is the special orthogonal group.

3. The torus with its diagonal removed, [math]\displaystyle{ S^1 \times \mathbb{R} }[/math], is the configuration space of two points on [math]\displaystyle{ S^1 }[/math]. This is [math]\displaystyle{ C_2(S^1) }[/math]

Reference: [1]