When you have a general coordinate chart on spacetime you have a lot of freedom to pick your coordinates, but you are always going to have 4 coordinates and each 4-tuple uniquely (in that chart) identifies one event in the manifold. When you are choosing generalized coordinates for a lagrangian the generalized coordinates represent degrees of freedom of your system. So you can have an arbitrary number of coordinates and each set of coordinates identifies a state of the system. Is there any sort of formal relationship between the two types of general coordinates? Sorry if the question is vague, I am not sure what I am looking for, but it kind of bothers me that the manifold coordinates are somewhat "special" compared to the lagrangian coordinates.