Coordinate Charts vs Generalized Coordinates

[Dale]

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.

 

[Fredrik]

I think that no matter how you choose your "generalized coordinates", they are given by a coordinate chart on a manifold called the configuration space of the system. The Lagrangian is a function from the tangent bundle of that manifold into the real numbers. The action functional takes curves in the configuration space to real numbers. See "Mathematical methods of classical mechanics" by V.I. Arnold, in particular chapter 4, pages 75-83.