Kirchhoff’s equations for the rigid body motion revisited

The purpose of the note is to relate Kirchhoff’s equations with Euler’s equations of rigid body dynamics and to establish the conservation laws arising from given external forces. This is done in vector form, for an inhomogeneous rigid body, by considering the kinetic energy and showing how the terms occurring in the Euler’s equations may be given the form of partial derivatives of the kinetic energy. Next it is shown how the vector form, via derivatives of the kinetic energy, provides the conservation laws and the corresponding first integrals along with their immediate mechanical meaning. An outline is given of different approaches in the literature to the derivation of Kirchhoff’s equations.