Intuitive

The configuration space or configuration manifold is the collection of all the possible "snapshots" or descriptions that the system can take.

Formulated differently, the configuration space is the possible "positions" of a mechanical system. Take note that the states of motion, eg. velocities/momenta are not part of the configuration space. This is in contrast to the phase space, where we also take the states of motion into account.

The state of a system is recorded in a configuration space point through all the locations and all the velocities that the objects in the system have at a given point in time.

The time evolution of a system can then be represented as a path in configuration space.

Concrete

Any set of parameters which can characterize the position of a mechanical system may be chosen as a suitable set of coordinates. They are called the "generalized coordinates" of the system and are the coordinates of the configuration manifold.

Examples

Free Particle

Free Particle on Space : it can be a 3 dimensional manifold, in fact, $\mathbb{R}^3$, and the coordinates can be multiple: Cartesian, spherical, cylindrical coordinates.

Multiple Free Particles

For 3 non-interacting point particles moving through space $\mathbb{R}^3$ the configuration space is $$\mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \cong \mathbb{R}^9$$

Each point in this space $\mathbb{R}^9$ corresponds to one specific configuration of our 3 particle system.

The 3 particles trace out continuous trajectories through $\mathbb{R}^3$, but the whole system can be said to trace out a continuous path through $\mathbb{R}^9$. So instead of three trajectories through $\mathbb{R}^3$ we now only have to track one trajectory through $\mathbb{R}^9$.

Rolling Sphere

Rolling Sphere on a Plane : the position of contact of the sphere on the plane, $\mathbb{R}^2$ and $SO(3)$ as the possible orientations of the sphere, so the configuration space is $\mathcal Q = \mathbb{R}^2 \times SO(3)$

Electromagnetism

For Electromagnetism , the thing is more triky, as the configuration space are not positions, but measures of the values of different fields over all space-time. The true coordinates of the fields could be the 4-potential $(\phi, A^1, A^2, A^3)$, so the configurations are 4 vector functions of 4 variables: $$ A: \mathbb{R}^4 \to \mathbb{R}^4; (x,y,z,t)\mapsto (\phi, A^1, A^2, A^3) $$ So $\mathcal Q = \mathcal{C}^2\{\mathbb{R}^4 \to \mathbb{R}^4\}$

References

1. [Lanczos, C.] Principles of Classical Mechanics

2. [Sussman G. J., Wisdom J. & Mayer M. E. ] Structure and Interpretation of Classical Mechanics

Abstract

As the name indicates, the mathematical figure is a manifold, and we will denote it as $\mathcal Q$. We will use the letter $q$ to denote the points of the system, and $q^i$ will denote the coordinates. The link between both is called configuration map $$ \chi:\mathcal Q \to \mathbb{R}^n; q \mapsto q^i=\chi^i(q) $$

The tangent bundle TQ will be referred to as configuration space, later on when we get to the chapter on Hamiltonian mechanics we’ll find a use for the cotangent bundle $T^∗Q$, and normally we call this the phase space.Lectures on Classical Mechanics, by John Baez

Lectures on Classical Mechanics, by John Baez

Why is it interesting?

In the Lagrangian formalism, we describe what happens in classical mechanics by referring to points in the configuration space. In this sense, the configuration space is for the Lagrangian formalism, what the phase space is for the Hamiltonian formalism.

Configuration spaces are also the central objects in robotics, as the set of reachable positions by a robot.