Rolling on an Arbitrary Surface


I found your excellent web site on rotational dynamics, and was wondering if you know of a reference that gives the complete set of differential equations of motion for a sphere or coin-shaped object rolling (without slipping) on an arbitrary continuous surface?

I was reading Dr. David Hestenes' book, "New Foundations for Classical Mechanics", which gives the combined differential equation, obtained by eliminating the reaction force terms in Newton's 2nd law and Euler's equation for a sphere rolling on a prescribed surface, and the equation of constraint (based on the no- slip condition at the point of contact). Dr. Hestenes mentions that the equation of constraint can be differentiated, and plugged back into the "combined" differential equation of motion, to get separate ODEs for dv/dt and dw/dt, where v = center of mass velocity, and w = angular velocity of the sphere.

Have you seen this combined ODE set, and do you know of a reference to them?


There is a slim volume titled "Mechanics" by Landau and Lifshitz published by Addison Wesley which has a section called "Rigid bodies in contact" where they discuss Lagrange's method and the d'Alembert principle for handling holonomic and non-holonomic constraints. I have not done much work in this area but this reference may be helpful. My edition of "Mechanics" is about 40 years old. I assume it is still in print. Check