What kind of manifold is a configuration manifold?

Question

I have recently been learning about the basic properties of topological, smooth, and Riemannian manifolds. But I frequently hear the term configuration manifold referenced in relation to Lagrangian mechanics.

What kind of manifold is a configuration manifold? Is this a mathematical concept or a physics concept?

Answer 1

It is not really a mathematical concept (as far as I know), but just a way of thinking.

Basically a manifold is a "configuration space" for a physical system if each point in the space corresponds to a "state" of the physical system, and "nearby" points represent "nearby" states.

For example, you could say that if you have a rod allowed to freely rotate in a sphere, where the two ends of the rod are indistinguishable, then the configuration space of this physical system is the real projective plane.

Answer 2

A phase space in classical mechanics might describe the values of position and speed (for instance) of a physical particle in some system. A configuration manifold is the manifold embedded in that phase space of physically allowable values of the particle's position and speed.