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I consider the equations of motion which follow from d'Alembert's principlefor a general mechanical system in a space of N dimensions, constrained by anon-holonomic constraint which is linear and homogeneous in the generalisedvelocities. The variational equations of motion which follow for the samesystem by assuming the validity of a specific variational action principle, inwhich the non-holonomic constraint is implemented by means of themultiplication rule in the calculus of variations are also considered. It isshown that these two types of equations of motion are not compatible in a spaceof dimension N greater than or equal to 3, if the constraint is genuinelynon-holonomic. This means that these two types of equations of motion do nothave coinciding general solutions.

On the compatibility of non-holonomic systems and certain related variational systems