Generalized vs curvilinear coordinates

Question

I am taking the course "Analytical Mechanics" (from on will be called "AM") this semester. In our first lecture, my professor introduced the notion of generalized coordinates. As he presented, we use generalized coordinates when we have constraints on the system (i.g. being on a sphere). When we have $M$ particles (in 3D space) we need to use $3M$ coordinates, unless we have $p$ independent constraints which now enable us to describe our system with $3M-p$ coordinates. We then use the transformation $\vec{r_i}=\vec{r_i}(q_1,q_2,...,q_N,t)$, and this is a transformation to a new coordinate system. This is where I got confused - in the course "Math For Physicists" (from now on will be called "MFP"), the professor said that for $n$ numbers to be a coordinate system, it needs to describe every point in space uniquely. But in the case I presented above (how my AM professor defined the notion of generalized coordinates) the coordinate system doesn't need to describe all the space, but rather the system (the position of the objects). Are the two notions do not refer to the same thing? i.e. is the concept of a generalized coordinate system which was introduced in the AM course different from the one of curvilinear coordinates that was introduced in the MFP course?

I think that the difference here is that while the generalized coordinates are indeed a form of curvilinear coordinates, they are not used for the same purpose and thus do not answer the demand that was presented in the MFP course.

Additionally, I would be thankful if you could provide a list of books or references where I could read about the formal definition of coordinate systems and curvilinear coordinates (the professor in MFP gave us references for books that are less rigorous and did not define or introduce curvilinear coordinates).

Answer 1

If you want to get to the heart of the matter, you need to understand the notion of a (smooth) manifold, whose definition will require you to understand what a (local) coordinate system/chart is.

• ‘generalized coordinates’ are just coordinate systems on a specific manifold in question (a so-called configuration manifold, which depends situation to situation).
• Curvilinear coordinates are sometimes used synonymously with the unprefaced ‘coordinate system/chart’. The Wikipedia link says they’re supposed to be coordinate systems on a Euclidean space… seems a little artificial to me, but whatever. I actually don’t like the way this page is written (way too narrowly written, so anytime you encounter a new situation, you’ll get confused again and wonder how things work), so I would actually just ignore it all together.

Anyway, don’t get hung up on the terminological distinctions; people use different names for the exact same thing, in different situations. The only reason things may seem ‘conceptually different’ is because you’re not learning the definitions sufficiently generally. Learn it once, and learn it generally, then everything will be easy. The whole purpose of coordinates is just to injectively assign (tuples of) numbers to given points, and the whole purpose of manifolds is to have a whole bunch of coordinate systems which ‘play nicely with each other’. See this MSE answer of mine for a more detailed description of what coordinates are, and just as importantly, what they are not. As for books, you just have to pick up any book on differential geometry, and read chapter 1 or 2.

Answer 2

I come from a background in robotics and in my mind, generalized coordinates are a subset of curvilinear coordinates.

What you want to describe is the kinematics of a system (all available motions) when using generalized coordinates. This is done by describing the relative motion relations between two objects, as a sequence of 1DOF motions, either pivots, sliding, or the more general screw motion. Each DOF presents itself as a generalized coordinate $q_i$ that might represent a rotation angle, translation, or combination.

These generalized coordinates are a subset of curvilinear coordinates because for mechanical systems they represent coordinate lines that are arcs, lines or helixes and curvilinear coordinates could be any curve.

Once the model space is described using the generalized coordinates, and any constraints also in the same coordinates you can proceed in the next step of using differential geometry to relate the kinematics to the dynamics of the system.

You have a lot of freedom on how to do this. There is no one way to do it correctly, but people gravitate towards model spaces that simplify the problem as much as possible.