Notes for AKT-140115/0:30:33

Configuration space Given a topological space $X$, the $n$th ordered configuration space of $X$ denoted by $\mathrm {Conf} _{n}(X)$ is the set of $n$-tuples of pairwise distinct points in $X$, that is $\mathrm {Conf} _{n}(X):=\prod ^{n}X\setminus \{(x_{1},\ldots ,x_{n}):x_{i}=x_{j}\;\mathrm {for} \;i\neq j\}$.

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 $\mathbb {R} ^{3}$ is $\mathbb {R} ^{3}$. For $n$ particles in $\mathbb {R} ^{3}$, it is $\mathbb {R} ^{3n}$

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

3. The torus with its diagonal removed, $S^{1}\times \mathbb {R}$, is the configuration space of two points on $S^{1}$. This is $C_{2}(S^{1})$

Reference: [1]