Configuration space (physics)

Not to be confused with PCI Configuration Space.

"C-space" redirects here. For the art gallery, see C-Space, Beijing.

Configuration space in physics

In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints. The configuration space of a typical system has the structure of a manifold; for this reason it is also called the configuration manifold.

For example, the configuration space of a single particle moving in ordinary Euclidean 3-space is just R³. For n particles the configuration space is R³ⁿ, or possibly the subspace where no two positions are equal. More generally, one can regard the configuration space of n particles moving in a manifold M as the function space Mⁿ.

To take account of both position and momenta one moves to the cotangent bundle of the configuration manifold. This larger manifold is called the phase space of the system. In short, a configuration space is typically "half" of (see Lagrangian distribution) a phase space that is constructed from a function space.

In quantum mechanics one formulation emphasises 'histories' as configurations.

Configuration spaces in mathematics

The configuration space of 2 not necessarily distinct points on the circle is the orbifold $T^{2}/S_{2},$ which is the Möbius strip.

In mathematics a configuration space refers to a broad family of constructions closely related to the state space notion in physics. The most common notion of configuration space in mathematics $C_{n}X$ is the set of n-element subsets of a topological space $X$. This set is given a topology by considering it as the quotient $C_{n}X=F_{n}X/\Sigma _{n}$ where $F_{n}X=\{(x_{1},\cdots ,x_{n})\in X^{n}:x_{i}\neq x_{j}\forall \ i\neq j\}$ where $\Sigma _{n}$ is the symmetric group acting by permuting the coordinates of $F_{n}X$. Typically, $C_{n}X$ is called the configuration space of n unordered points in $X$ and $F_{n}X$ is called the configuration space of n ordered or coloured points in $X$; the space of n ordered not necessarily distinct points is simply $X^{n}.$

If the original space is a manifold, the configuration space of distinct, unordered points is also a manifold, while the configuration space of not necessarily distinct unordered points is instead an orbifold.

Configuration spaces are related to braid theory, where the braid group is considered as the fundamental group of the space $C_{n}{\mathbb {R} }^{2}$.

A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle $\pi \colon E_{n}\to C_{n}$ which is a subbundle of the trivial bundle $C_{n}\times X^{n}\to C_{n}$, and which has the property that the fiber over each point $p\in C_{n}$ is the n element subset of $X_{n}$ classified by p.

The homotopy type of configuration spaces is not homotopy invariant – for example, two lens spaces that are homotopy equivalent but not homeomorphic (in this case meaning simple homotopy equivalent) need not have homotopy equivalent configuration spaces, which can be detected by Massey products.

See also

• Feature space (topic in pattern recognition)
• Parameter space
• Phase space
• State space (physics)

External links

• Intuitive Explanation of Classical Configuration Spaces.
• Interactive Visualization of the C-space for a Robot Arm with Two Rotational Links from UC Berkeley.