Configuration space
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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 were equal. More generally, one can regarded 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 are related to braid theory, also, since the condition on a string of not passing through itself is formulated by cutting diagonals out of function spaces.