Approximation of Infinitesimal Rotations in Calculus of Variations

A consistent and comprehensive first-order approximation method was presented for dealing with infinitesimal angular displacements in dynamic problems in which variational methods are being used to derive the equations of motion. The presented approach has been employed to derive the correct and complete (first-order) dynamic equations of a symmetric spinning top and a gyroelastic continuum. It has been shown that, since the infinitesimal rotations do not obey the parallelogram addition law to second order, using a second-order approximation on infinitesimal rotations while treating them as Euclidean vectors is discrepant and should be avoided.