Hallo, I\'m Giorgio. I\'ve read your solution to Goldstein\'s Classical Mechanic

Question

Hallo, I'm Giorgio. I've read your solution to Goldstein's Classical Mechanichs exercise 5.24 'A wheel rolls down a flat inclined surface ...." about undetermined Lagrangian multipliers. In this exercise the wheel "may rotate about the axis normal to the surface" and I can find in the figure angular twisting variation , but no kinetic energy related in the Lagrangian and the final solution is just similar to example (not twisting) in Goldstein's figure 2.5 paragraph 2.4 "Extensions of Hamilton's Principle to nonholonomic Systems" .

Can you help me to wrightly understand the tems of this exercise?

Many thanks,

Giorgio.

Explanation / Answer

"may rotate about the axis normal to the surface"

This means that suppose the surface is the plane of the paper. Then the axis perpendicular to the surface will be coming out of the plane of paper. So, it is this axis about which the wheel can rotate.

Hamilton's principle

Hamilton's principle states that the development in time for a mechanical system is such that the integral of the difference between the kinetic and the potential energy is stationary

Nonholonomic system.

A nonholonomic system in physics and mathematics is a system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state.

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