Holonomic constraints and degrees of freedom

Question

Wikipedia and other sources define holonomic constraints as a function

$$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$

and says the number of degrees of freedom in a system is reduced by the number of independent holonomic constraints.

I could take multiple such constraints $f_1, \ldots, f_m$ and formulate them as single one that is fulfilled if and only if all $f_i$ are fulfilled:

$$ f = \sum_{i=1}^{m}{\lvert f_i \rvert}. $$

This combined $f$ would obviously reduce the number of degrees of freedom by $m$ instead of $1$.

Alternatively, to avoid the absolute value, I could use a sum of squares

$$ f = \sum_{i=1}^{m} f_i^2 $$

instead. Where is my error in reasoning?

Answer 1

Well, in the definition of holonomic constraints $f_1, \ldots, f_m$, there are also two technical regularity conditions (which OP's counterexamples do not fulfill):

1. The functions $f_1, \ldots, f_m,$ should be continuously differentiable with $m\leq 3N$.

2. The $m\times 3N$ rectangular Jacobian matrix $$\frac{\partial(f_1, \ldots, f_m)}{\partial(\vec{r}_1, \ldots, \vec{r}_N)}$$ should have rank $m$ on the constraint submanifold.

The regularity conditions 1 & 2 are imposed to ensure the local existence of generalized coordinates $q_1, \ldots, q_n$, in some open neighborhood, where $n:=3N-m$, via the inverse function theorem.

See also this related Phys.SE post.

References:

1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; Subsection 1.1.2, p. 7.