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Rolling Cylinder

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A cylinder of mass m and radius a is rolling on a stationary larger cylinder of radius R.
At what angle will the smaller cylinder will leave the surface and what is the friction force between the cylinders (assume coefficient mu).

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Solution Summary

The solution uses Lagrangian mechanics to solve all the equations of motion and show how to use constraints in conjunction with the standard Lagrangian.
The solution is 6 pages long with full derivations and appendix.

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See attached files.

The energy of the motion of the cylinder's center of mass is:


The energy of the cylinder's rotation about its own center of mass is:

The potential energy is:


Thus the general Lagrangian is:


Now we get to the constraints that are imposed on the coordinates.

The first constraint is the fact that the cylinder rolls on the surface of the lower cylinder:


Or this can be written as a homonymic constraint

The generalized force in the direction of the r coordinate is (see appendix):

This is a force that is applied on the small cylinder by the large cylinder in the radial direction. In ...

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