Configuration space

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• Konfigurationsraum
• Espacio_de_configuración
• Espace_de_configuration
• Configuratieruimte
• Espaço_de_configuração
• 位形空间

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"Configuration space" may also refer to PCI Configuration Space.

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 R^3N, 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 M^N.

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

In mathematics a configuration space refers to a broad family of constructions closely related to the state space notion in physics (see above). The most common notion of configuration space in mathematics C_nX 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_nX = F_nX / Σ_n where where Σ_n is the symmetric group acting by permuting the coordinates of F_nX. Typically, C_nX is called the configuration space of n unordered points in X and F_nX is called the configuration space of n ordered or coloured points in X.

Configuration spaces are related to braid theory, where the Braid group is considered as the fundamental group of the space .

See also

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

External links

• [1] Intuitive explanation of classical configuration spaces.
• [2] interactive visualization of the C-space for a robot arm with two rotational links from the university of Berkeley.