The lagrange multipliers associated with the rotation matrix characterizing the motion of a rigid body about its centre of mass

Abstract To describe the motion of a rigid body, parametrization based on the use of a rotation matrix consisting of nine components is chosen instead of angular parameters. The equations of motion of mechanical systems consisting of many bodies coupled to one another turn out to be linear. The description of the rotations is provided by six Lagrange multipliers, grouped in a symmetrical 3 × 3 matrix, denoted by Λ, the components of which are related to the volume averages of the internal couplings in the body. The following properties are proved for a rigid body rotating about its centre of mass: the negative of the Lagrange multiplier matrix is positive, and at each instant of time an orthonormalized basis exists in which new components of the matrix Λ are constant, which gives six first integrals of the equations of motion [1]. It is proved that three eigenvalues of the matrix Λ do not change with time and, moreover, they can be found in explicit form.