Example: a particle in 3D space[edit]
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.
Example: rigid body in 3D space[edit]
The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted where represents the coordinates of the origin of the frame attached to the body, and represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from and three from , and is said to have six degrees of freedom.
In this case, the configuration space is six-dimensional, and a point is just a point in that space. The "location" of in that configuration space is described using generalized coordinates; thus, three of the coordinates might describe the location of the center of mass of the rigid body, while three more might be the Euler angles describing its orientation. There is no canonical choice of coordinates; one could also choose some tip or endpoint of the rigid body, instead of its center of mass; one might choose to use quaternions instead of Euler angles, and so on. However, the parameterization does not change the mechanical characteristics of the system; all of the different parameterizations ultimately describe the same (six-dimensional) manifold, the same set of possible positions and orientations.
Some parameterizations are easier to work with than others, and many important statements can be made by working in a coordinate-free fashion. Examples of coordinate-free statements are that the tangent space corresponds to the velocities of the points , while the cotangent space corresponds to momenta. (Velocities and momenta can be connected; for the most general, abstract case, this is done with the rather abstract notion of the tautological one-form.)
Formal definition[edit]
In classical mechanics, the configuration of a system refers to the position of all constituent point particles of the system.[2]
Phase space[edit]
The configuration space is insufficient to completely describe a mechanical system: it fails to take into account velocities. The set of velocities available to a system defines a plane tangent to the configuration manifold of the system. At a point , that tangent plane is denoted by . Momentum vectors are linear functionals of the tangent plane, known as cotangent vectors; for a point , that cotangent plane is denoted by . The set of positions and momenta of a mechanical system forms the cotangent bundle of the configuration manifold . This larger manifold is called the phase space of the system.
