Configuration-space manifolds

In advanced classical mechanics (CM), a system under observation (SUO) is modelled by a single point, the system point, moving in some real r-dimensional space known as configuration space. Typically, configuration spaces are real differentiable manifolds chosen to model the mechanical degrees of freedom believed to represent the SUO.

Whilst it is always more elegant to discuss problems in CM over manifolds in a coordinate-free way (Abraham and Marsden, 2008), it is generally more convenient to use coordinates and we shall do so here. Therefore, having chosen a configuration-space manifold C relevant to the problem at hand, the next step is to set up a system of coordinate frames or patches to cover the region of interest in the manifold.

In CM, the region of interest will contain the trajectory of the system point over the time interval of interest. Over a finite interval of time, a system point will not move in general over the whole of its configuration space. In practice, therefore, the equations of motion can be discussed relative to a single coordinate patch, which need not be global. This is fortunate for CM, because for many manifolds, such as spheres, a single well-behaved coordinate patch covering the whole manifold cannot be constructed.