1
$\begingroup$

enter image description here

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?

$\endgroup$
  • 1
    $\begingroup$ 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 at 20:02
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.