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This particular question is from eq. (1.39) in Goldstein "Classical mechanics".

These are the constraint equations

I've seen 2 kinds of solutions for a pure rolling disc on a 2D plane (i) using "differential 1-form" and (ii) using "vector field props" (curl specifically) both of which I can't quite grasp properly. So looking for some human answers with insights.

Qmechanic
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Gowriprakash
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1 Answers1

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  • OP is essentially asking why a ball or disk rolling$^1$ on a 2D plane is a semi-holonomic rather than a holonomic system?

  • In contrast, e.g. a cylinder rolling on a 2D plane is a holonomic system.

Here is an intuitive realization of these notions using no math:

  • On one hand, if one draws a $\color{green}{\rm green}$ spot on the cylinder and a $\color{red}{\rm red}$ spot on the 2D plane, and assemble the system by making the 2 spots touch, then after arbitrary rolling (possibly back and forth), every time the $\color{red}{\rm red}$ spot touches the cylinder, it also touches the $\color{green}{\rm green}$ spot.

  • On the other hand, for the analogous experiment with a ball or disk, this property of the $\color{green}{\rm green}$ and $\color{red}{\rm red}$ spots is no longer necessarily true. This shows that the rolling constraint is not described purely by positions, i.e. it is non-holonomic.

References:

  1. H. Goldstein, Classical Mechanics; Eq. (1.39) & Fig. 1.5.

$^1$ All rolling in this answer is pure rolling without slipping.

Qmechanic
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