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I'm reading Goldstein's Classical Mechanics and he defines a constraint on particles having radii $\mathbf{r}_i$ to be holonomic if it can be written as $f(\mathbf{r}_1, \mathbf{r}_2, \dots , t) = 0$. He then says that an example of a non-holonomic constraint would be particles constrained to the exterior of a sphere: $r^2 - a^2 \ge 0$.

However, this could be rewritten as $\mathbf{1}_{\left\{\mathbf{r}\mid \:|r|^2 < a^2\right\}}(\mathbf{r}) = 0$, where $\mathbf{1}$ denotes an indicator function. As such it seems to me that the given example is a holonomic constraint. What am I missing?

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  • $\begingroup$ I am not used to the indicator function, but can it be exactly written as $f(\mathbf{r}_1, \mathbf{r}_2, \dots , t) = 0$? If not, then the constraint is not holonomic. $\endgroup$ – Diracology Jul 15 '17 at 0:18
  • $\begingroup$ The indicator function is $f$ in this case. It takes a value of 1 when the coordinates are in the set of points which violate the constraint. (And 0 otherwise) $\endgroup$ – Davis Yoshida Jul 15 '17 at 0:41
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    $\begingroup$ I think to be holonomic $f$ should also be differentiable, and the indicator function is not. $\endgroup$ – hyportnex Jul 15 '17 at 1:06
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    $\begingroup$ Related: physics.stackexchange.com/q/310459/2451 $\endgroup$ – Qmechanic Jul 15 '17 at 2:08