For an unconstraint system described by N number of particles, there are 3N independent generalised coordinates; and in fact, 3N degrees of freedom for this system.
If the system were constraint by an equation of motion given by some function
$G=G\left ( x_{1},y_{1},z_{1},\cdot \cdot \cdot ,x_{N},y_{N},z_{N} \right )$
then, the degree of freedom for this constrained system is now reduced to 3N-1.
However, my notes at times employ the term "independent equation of constraint". What differences exists between the usage "independent equation of constraint" and "equation of constraint"? What does this mean for "S independent equation of constraint" for a system of N particles?
Clarification is appreciated.
Thanks in advance.
Suppose we have two variables, $x$ and $y$.
I tell you that $$ G(x, y) = 0 $$ where $$ G(x, y) = x-y. $$ That's a constraint, and reduces the number of independent variables. Now if I also tell you that $$ H(x, y) = 0, $$ you might say that again I've reduced the number of independent variables. But that's not true. For instance, if $$ H(x, y) = (x-y)^2 $$ then the solutions to $H = 0$ are exactly the same as those of $G(x, y) = 0$. If $H$ was $\sin(x-y)$, then the solutions of $H=0$ are a superset of the solutions of $G = 0$. Both of these are instances of "not independent."
I don't know how your book defines things, but in general, an equation like $G = 0$ (for some smooth enough function $G$) defines a submanifold (unless $\nabla G (P) = 0$ for some point $P$ with $G(P) = 0$, in which case all bets are off). [That's just the implicit function theorem.]
A second equation, $H = 0$, defines another submanifold. And I'd say that $G$ and $H$ are "independent" if these two submanifolds intersect transversly, which amounts to saying that at any point $Q$ of the intersection, the set $\{\nabla G(Q), \nabla H(Q)\}$ is linearly indepdent. The generalization to further constraints is similar.
(I don't know whether my definition is what the physics people say, because, like you, I have some difficulty reading physics texts.)