How to determine whether a set of coordinates are independent and sufficient to determine the system completely?

Question

In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates $q_j$s they are all independent each other in a way that they implicitly contain the constraints, so that changing the value of one of $q_i$s does not force force us to change the other $q_j$s, i.e they are independent.

However, in none of the books or lectures that I've seen, there haven't been anything mentioned about how to find those independent coordinates, or any thing about the issue that if we have some set of coordinates, how we can test whether they are all independent and a complete set of coordinates s.t they describe the system completely, or even the number of independent coordinates it is needed.

Of course, there are lots of things talked about the degrees of freedom of a system, but I haven't seen anything about how to determine the degrees of freedom of a given system.

Answer 1

Though the question sounds like a Classical Mechanics problem, it's actually linear algebra. But I learnt it from CM texts. And I don't have answers to all the questions from your second paragraph. So, I'm just writing what I know. Actually, just rephrasing the words from the books mentioned in the reference. Suppose that we've a system of $N$ particles, subjected to $k$ constraints of the form $$f_l(r,t)=0; \space\space\space\space\space\space\space\space l=1,2,...,k $$ where $r_i$'s are our Cartesian coordinates, with $i=1,2,...,n$. Now from linear algebra we can say,

The constraints $f_l=0$ are independent if the $k\times n $ matrix with entries $\partial f_l \over\partial r_i$ has miximal rank at each point.

Here's the first condition: The constraints $f_l=0$ must be independent. Independence precludes meaningless constraints (such as t=0) and prevents double counting. It fails, if $f_m=f_n$ for some $m\neq n$.$^1$
Now, consider the transformation $$q_j=q_j(r_i); \space\space\space\space\space\space\space\space\space j=1,2,...,n-k.$$ If the matrix $[{\partial r_i \over \partial q_j}]_{n\times (n-k)}$ has the maximum rank, then the quoted statement guarantees the independence of $q_j$'s, with $j=1,2,..., n-k$. But we don't know yet if they form generalized coordinates.
If the vectors $\partial \mathbf r \over \partial q_j$ are orthogonal to $\nabla_{r_i} f_l$, they form the basis for the configuration space and $q_j$'s form generalized coordinates.$^2$

---

$^1$ N. M. J. Woodhouse, Introduction to Analytical Dynamics, Springer-Verlag, London, 2009.
$^2$ Antonio Fasano and Stefano Marmi, Analytical Mechanics: An Introduction, Oxford University Press, 2006.