Are generalized coordinates, as used in Legrangian mechanics, just a different name for coordinates on a chart in a manifold? The idea of generalized coordinates never quite "clicked" with me, but after reading a paper today, it seems that they are just an implicit way of working with manifolds. So, say you're working with a double-pendulumn system. You have two objects (the ends of the pendulumns) in 3D space, for a total of 6 degrees of freedom, so the entire system can be modeled in R^6. But, to simplify the math, we can use our knowledge of the system's invariants to reduce this down to a 2D submanifold, parametrized by the angle of each pendulumn as each swings in a plane. Am I correct in coming to this conclusion?