# Holonomic constraints and degrees of freedom

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?

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.