Rolling on an Arbitrary Surface

## Question:

hello

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?

## Answer:

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 Amazon.com.