I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates? the way I got it, every coordinate system different from the standard cartesian-one is a set of generalized coordinates...?
Generalized coordinates refer to any coordinate system. i.e. a statement about generalized coordinates holds for cartesian, spherical, cylindrical, etc. coordinate systems. In particular, one is free to choose any convenient coordinate system for a problem and solve the problem using Lagrange's equations for that coordinate system.
Yes, polar and spherical coordinates are generalized coordiantes for the position of a single particle. But general coordinates are a lot moe general. And cartesian coordinates are, technically at least, also "general coordinates". Carl