Within Newtonian mechanics, the logic is usually the opposite: We are given $N$ point-particle positions in rectangular 3-space $\mathbb{R}^3$. They satisfies $3N-n$ holonomic constraints, so that we can (at least locally) define $n$ generalized coordinates.

Within Newtonian mechanics, the logic is usually the opposite: We are given $N$ point-particle positions in rectangular 3-space $\mathbb{R}^3$. They satisfies $3N-n$ holonomic constraints, so that we can (at least locally) define $n$ generalized coordinates.

References:

1. H. Goldstein, Classical Mechanics; Chapter 1.