Editing Configuration space (physics) (section)

===Rigid body in 3D space===
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 <math>\mathbb{R}^{3}\times\mathrm{SO}(3)</math> where <math>\mathbb{R}^{3}</math> represents the coordinates of the origin of the frame attached to the body, and <math>\mathrm{SO}(3)</math> 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 <math>\mathbb{R}^{3}</math> and three from <math>\mathrm{SO}(3)</math>, and is said to have six [[Degrees of freedom (mechanics)|degrees of freedom]].

In this case, the configuration space <math>Q=\mathbb{R}^{3}\times\mathrm{SO}(3)</math> is six-dimensional, and a point <math>q\in Q</math> is just a point in that space. The "location" of <math>q</math> 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 angle]]s 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 [[quaternion]]s 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]] <math>TQ</math> corresponds to the velocities of the points <math>q\in Q</math>, while the [[cotangent space]] <math>T^*Q</math> 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]].)