Physics Asked by Christian Schnorr on July 18, 2021
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):
The functions $f_1, ldots, f_m,$ should be continuously differentiable with $mleq 3N$.
The $mtimes 3N$ 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:
Answered by Qmechanic on July 18, 2021
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