gauge theory - primary constraints for constrained Hamiltonian systems

I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian systems" by H. Rothe and K. Rothe, and "Quantization of gauge system" by Henneaux and Teitelboim.

Consider a system with Lagrangian $L(q,\dot{q})$ and define the momenta $p_j = \partial L/\partial{\dot{q}_j}$, with $j$ from 1 to $n$ (for $n$ degrees of freedom), and the Hessian $W_{ij}(q,\dot{q}) = \partial^2 L/\partial{\dot{q}_i}\partial{\dot{q}_j}$. Let $R_W$ be the rank of the Hessian $W_{ij}$.

Assume that $L$ is singular and $R_W < n$. In order to ask the question, I'm now quoting how the Rothes describe the setting of primary constraints on pages 26 and 27 of their book.

Let $W_{ab}$, ($a,b = 1,...,R_W$) be the largest invertible submatrix of $W_{ij}$, where a suitable rearrangement of the components has been carried out. We can then solve the eqs. $p_j = \partial L(q,\dot{q})/\partial{\dot{q}_j}$ (1) for $R_W$ velocities $\dot{q}_a$ in terms of the coordinates $q_i$, the momenta $\{p_a\}$ and the remaining velocities $\{\dot{q}_\alpha\}$: $\dot{q}_a = f_a(q,\{p_b\},\{\dot{q}_\beta\})$, with $a,b = 1,...,R_W$ and $\beta = R_W + 1,...,n$.

Inserting this expression into the definition of $p_j$, one arrives at a relation of the form $p_j = h_j(q,\{p_a\},\{\dot{q}_\alpha\})$. For $j = a$ ($a = 1,...,R_W$) this relation must reduce to an identity. The remaining equations read $p_\alpha = h_\alpha (q,\{p_a\},\{\dot{q}_\beta\})$. But the rhs cannot depend on the velocities $\dot{q}_\beta$, since otherwise we could express more velocities from the set $\{\dot{q}_\alpha\}$ in terms of the coordinates, the momenta, and the remaining velocities.

This is where the presentation from Rothe stops, and my concern is that equations of the form $p_\alpha = h_\alpha (q, \{p_a\},\{\dot{q}_\beta\})$ (2) with all $\{\dot{q}_\beta\}$ present can still be possible, and yet one cannot solve for more velocities from the set $\{\dot{q}_\alpha\}$ in terms of the coordinates, the momenta and the remaining velocities if the conditions stipulated in the Implicit Function Theorem are not met, for not all equations of the type (2) can be solved implicitly for $\{\dot{q}_\alpha\}$. Therefore it is not proven that there are $(n - R_W)$ primary constraints of the form $\phi_\alpha (q,p) = 0$.

Henneaux and Teteilboim even state that these $(n - R_W)$ constraints of the form $\phi_\alpha (q,p) = 0$ are functionally independent, but give no justification to this statement.

I would be most thankful if you could help clarify my above concern and also if you could clarify the statement by Henneaux and Teitelboin as to the fact that the constraints are functionally independent. Thank you!

Answer

OP wrote

This is where the presentation from Rothe stops, and my concern is that equations of the form $p_\alpha = h_\alpha (q, \{p_a\},\{\dot{q}_\beta\})$ (2) with all $\{\dot{q}_\beta\}$ present can still be possible, and yet one cannot solve for more velocities from the set $\{\dot{q}_\alpha\}$ in terms of the coordinates, the momenta and the remaining velocities if the conditions stipulated in the Implicit Function Theorem are not met, for not all equations of the type (2) can be solved implicitly for $\{\dot{q}_\alpha\}$. Therefore it is not proven that there are $(n - R_W)$ primary constraints of the form $\phi_\alpha (q,p) = 0$

I sensed a misunderstanding of Rothe's statements, ignore me if I'm not understanding OP correctly:

Rothe is arguing at least one of the $\dot{q}_\beta$'s can be expressed as a function of $p_a,p_\alpha$ and remaining $\dot{q}_\alpha$'s. For any particular $\alpha$ in your equation (2), by implicit function theorem applied to one-variable($\dot{q}_\beta$) function, it's always doable unless $\frac{\partial h_\alpha}{\partial \dot{q}_\beta}=0$ for our chosen $\alpha$ and $\beta$, but the latter case simply means $h_\alpha$ does not depend on $\dot{q}_\beta$, so either case Rothe's statement is correct.

Updates:

As for the functional independence part, if I'm not mistaken, is quite trivial. It's because in your equation (2), all the $p_\alpha$'s are on LHS and RHS doesn't contain any $p_\alpha$ thus it's impossible to find a inter-relation among these equations(plural in the sense that $\alpha$ can take many values, from $1$ to $M'$). Or in differential calculus language, the constraint functions from (2) will be $\phi_\alpha(q,p)=p_\alpha-h_\alpha(q,\{p_a\})$, thus the Jacobian of these functions will simply be

$\frac{\partial \phi_\beta}{\partial \{p_\alpha,p_a, q\}}= \begin{bmatrix} 1 & 0 & \cdots &0&\cdots&\frac{\partial h_1}{\partial p_a}&\cdots&\frac{\partial h_1}{\partial q_i}&\cdots\\ 0 & 1 & \cdots&0&\cdots&\frac{\partial h_2}{\partial p_a}&\cdots&\frac{\partial h_2}{\partial q_i}&\cdots\\ \vdots & \vdots & \ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0 &0 &\cdots&1&\cdots&\frac{\partial h_{M'}}{\partial p_a}&\cdots&\frac{\partial h_{M'}}{\partial q_i}&\cdots\end{bmatrix}$

And this Jacobian is of maximal rank because of the identity submatrix on the left, and this is the same as saying these contraint functions are functionally independent.

P.S.: Now I'm not quite sure what situation H&T were referring to when they said $M