Editing Configuration space (physics) (section)

===A particle in 3D space===
The position of a single particle moving in ordinary [[Euclidean space|Euclidean 3-space]] is defined by the vector <math>q=(x,y,z)</math>, and therefore its ''configuration space'' is <math>Q=\mathbb{R}^3</math>.  It is conventional to use the symbol <math>q</math> for a point in configuration space; this is the convention in both the [[Hamiltonian mechanics|Hamiltonian formulation of classical mechanics]], and in [[Lagrangian mechanics]]. The symbol <math>p</math> is used to denote momenta; the symbol <math>\dot{q}=dq/dt</math> refers to velocities.

A particle might be constrained to move on a specific [[manifold]]. For example, if the particle is attached to a rigid linkage, free to swing about the origin, it is effectively constrained to lie on a sphere. Its configuration space is the subset of coordinates in <math>\mathbb{R}^3</math> that define points on the sphere <math>S^2</math>. In this case, one says that the manifold <math>Q</math> is the sphere, ''i.e.'' <math>Q=S^2</math>.

For ''n'' disconnected, non-interacting point particles, the configuration space is <math>\mathbb{R}^{3n}</math>. In general, however, one is interested in the case where the particles interact: for example, they are specific locations in some assembly of gears, pulleys, rolling balls, ''etc.'' often constrained to move without slipping. In this case, the configuration space is not all of <math>\mathbb{R}^{3n}</math>, but the subspace (submanifold) of allowable positions that the points can take.