nLab
configuration space

Idea

The configuration space of a physical system is the space of all possible states of a physical system. For a classical mechanical system with $n$-degrees of freedom it is a manifold of dimension $n$.

For some purposes, like in relativistic field theory it is convenient to consider not space of states at the point in time, but rather the space of histories or trajectories.

In mathematics, one often talks about the configuration space of $N$ distinct points in a manifold $M$, see Fadell's configuration space.

Idea of histories variant

For a system described by an action functional the configuration space is the domain space of that functional.

More precisely, let $H$ be the ambient (∞,1)-topos with a natural numbers object and equipped with a line object ${𝔸}^1$. Then an action functional is a morphism

$$\exp (iS(-)):C\to {𝔸}^1/\mathbb{Z}$$\exp(i S(-)) : C \to \mathbb{A}^1/\mathbb{Z}

in $H$. The configuration space is the domain $C$ of this functional.

If $H$ is a cohesive (∞,1)-topos then there is an intrinsic differential of the action functional to a morphism

$$d\exp (iS(-)):C\to {♭}_{\mathrm{dR}}{𝔸}^1/\mathbb{Z}.$$d \exp(i S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z} \,.

This is the Euler-Lagrange equation of the system. The homotopy fiber

$$P\to C$$P \to C

of this morphism is the covariant phase space inside the configuration space: the space of classically realized trajectories/histories of the system.