On Hamel’s paradox

Abstract

In his book Hamel pointed out through an example that the embedding of a nonholonomic constraint directly in the Lagrangian of an unconstrained mechanical system causes one to obtain incorrect equations of motion for the constrained system upon the application of Lagrange’s formalism. Wanichanon and Udwadia provided the reason for this and illustrated their result through a series of examples. They also gave the reason why such an embedding for holonomic constraints in the Lagrangian gives the correct equations of motion, a view conjectured by Rosenberg in his book. A recent paper by Ye-Hwa Chen again raised the issue of Hamel’s Paradox and claimed that embedding holonomic constraints in the Lagrangian of an unconstrained mechanical system can yield, in general, incorrect equations of motion. The purpose of this paper is to provide a resolution to the problem of whether the embedding of holonomic constraints in the Lagrangian yields the correct equations of motion of a constrained mechanical system. It is shown that the complete embedding of honolomic constraints in the Lagrangian when used properly with the Lagrange formalism will yield the correct equations of motion of a mechanical system.