Question about Lagrangian mechanics I have a question about the Lagrangian-formalism; they state that the holonomic constraints are expressed as:
$$f_\alpha(x_1,y_1,z_1,...,x_n,y_n,z_n,t) = 0 $$ where $\alpha = 1,...,L$ 
They state now that these constraints may not contain irrelevant information. They express this by saying:
$$rank(\frac{\partial{f^{\alpha}}}{\partial{x_k}}\frac{\partial{f^{\alpha}}}{\partial{y_k}}\frac{\partial{f^{\alpha}}}{\partial{z_k}}) = L$$
With $\alpha = 1,..,L $ and $k = 1,...,N $.
This looks like a Jacobi-matrix.   I don't understand however how this expresses that the constraints may not contain irrelevant information. (I see that they may not be linear dependent, is there a relation between linear dependence and the Jacobi-matrix?)
Further they state that under these premises, we can solve the constraints and we can say:
$z_k = z_k(x_1,y_1,z_1,...,t) $ (where $z_k$ is a symbolical notation for a parameter (so it can also be for instance $x_5$)), and $z_k$ depends on $3n-L$ parameters and the time $t$. With a regular transformation these parameters can be transformed to other generalized parameters. This should follow out of the theorem of implicit functions, but these theorem only states that we can solve $f_\alpha$ local: so I don't understand why they assume that these generalized parameters should completely determine the configuration of the system. It seems that these coordinates only determine the configuration of the system in the surrounding of a point that satisfies the constraints. 
 A: Implicit function theorem states that a relation can be transformed in a function, that is from the relation:
$$f(x_1,y_1,z_1,........x_N,y_N,z_N, t) = 0$$ 
You could express $x_d$ (for instance) as a function:
$x_d = x_d ((x_1,y_1,z_1,...,x_{d -1},y_{d -1},z_{d -1}, y_d,z_d,x_{d +1},y_{d +1},z_{d +1}.....x_N,y_N,z_N, t))$
The rank of the Jacobi Matrix indicates the number of  linearly independent columns ( or linearly independent rows), This means that the holonomic constraints are independent equations, that is : there are really $L$ different constraints.
Furthermore, considering space variables,you have $3N$ variables $x_1,y_1,z_1,........x_N,y_N,z_N$, with $L$  true constraints, so in fact, you have only 
$m = 3N - L$ freedom degrees for space variables. 
Adding time, we see that you have $m = 3N - L$ freedom degrees for space variables and one freedom degree for time.
Setting $q_1, q_2, .........q_m$ for the space freedom degrees, this means that every $x_d$ or $y_d$ or $z_d$ could be written as a function of $q_1, q_2, .........q_m$ and $t$
$x_d = x_i(q_1, q_2, .........q_m, t)$
