Configuration space
Encyclopedia
Configuration space in physics
In classical mechanicsClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the configuration space is the space of possible positions that a physical system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
may attain, possibly subject to external constraints. The configuration space of a typical system has the structure of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
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
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
Mn.
To take account of both position and momenta one moves to the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
of the configuration manifold. This larger manifold is called the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of the system. In short, a configuration space is typically "half" of (see Lagrangian distribution) a phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
that is constructed from a function space.
In quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
one formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
emphasises 'histories' as configurations.
Configuration spaces in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a configuration space refers to a broad family of constructions closely related to the state space
State space
In the theory of discrete dynamical systems, a state space is a directed graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ = b where the function f defines the dynamical system.State spaces are...
notion in physics. The most common notion of configuration space in mathematics
is the set of n-element subsets of a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. This set is given a topologyTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
by considering it as the quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
where 
where 
is the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
acting by permuting the coordinates of
. Typically, 
is called the configuration space of n unordered points in 
and 
is called the configuration space of n ordered or coloured points in 
; the space of n ordered not necessarily distinct points is simply 
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
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
.
Configuration spaces are related to braid theory
Braid theory
In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...
, where the braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
is considered as the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the space
.A configuration space is a type of classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
or (fine) moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
. In particular, there is a universal bundle
which is a subbundle of the trivial bundle 
, and which has the property that the fiber over each point 
is the n element subset of 
classified by p.The homotopy type of configuration spaces is not homotopy invariant – for example, two lens space
Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....
s 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 product
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple product:...
s.
See also
- Feature spaceFeature spaceIn pattern recognition a feature space is an abstract space where each pattern sample is represented as a point in n-dimensional space. Its dimension is determined by the number of features used to describe the patterns...
(topic in pattern recognition) - Parameter spaceParameter spaceIn science, a parameter space is the set of values of parameters encountered in a particular mathematical model. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function....
- Phase spacePhase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
- State space (physics)State space (physics)In physics, a state space is a complex Hilbert space within which the possible instantaneous states of the system may be described by a unit vector. These state vectors, using Dirac's bra-ket notation, can often be treated as vectors and operated on using the rules of linear algebra...