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From the constraint $v=a\dot{\phi}$ of a rolling disk over a plane, where $a$ is the radius of the disk we can derive these two equations:

we have two differential equations of constraint:

$dx=asin\theta d\phi$

$dy=-acos\theta d\phi$

Can you rigorously explain to me why can't I integrate these functions? Is it their non-integrability that makes these constraints non-holonomic?

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    $\begingroup$ Note that this question is Derivation 4 in Chapter 1 of Goldstein's Classical Mechanics, from which your example, figure, and notation are taken. $\endgroup$ – Michael Seifert May 28 '19 at 20:02

HINT: Rewrite the first constraint as $$ f \left[ dx +(- a \sin \theta) d\phi + (0) d\theta \right] = 0. $$ where $f(x, \theta, \phi)$ is some unknown integrating function. We want to know whether this can be written as $$ dg = \frac{\partial g}{\partial x} dx + \frac{\partial g}{\partial \phi} d\phi+ \frac{\partial g}{\partial \theta} d\theta $$ Assuming that $g$ is a "nicely-behaved" function of the coordinates, its mixed partial derivatives are independent of the order of differentiation. Using this fact, can you show that this implies $f = 0$?

To answer your second question: there are different definitions of "non-holonomic" constraints used by different authors. If you're following Goldstein, any constraint that cannot be written in the form $f(x_i) = 0$ is non-holonomic; but there are also non-holonomic constraints that can only be written as equalities among the higher derivatives, or as inequalities.

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