Configuration space (physics)

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed. The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector , and therefore its configuration space is . It is conventional to use the symbol for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian mechanics. The symbol is used to denote momenta; the symbol 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 that define points on the sphere . In this case, one says that the manifold is the sphere, i.e. . For n disconnected, non-interacting point particles, the configuration space is . 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 , but the subspace (submanifold) of allowable positions that the points can take.