Integration of the equations of a rotational motion of a rigid body in quaternion algebra. The Euler case

Abstract

A dynamical system is constructed in the multiplicative group of the quarternion algebra H that serves as the configuration space. A homomorphism H → SO(3) is used such that the unit sphere S 3 ⊂ H, invariant under the system, is transformed into the rotation group SO(3). The homomorphic image of the system is identical with the dynamics of rotational motion of a rigid body. The equations of motion are completely integrated in the Euler case. To this end Weierstrass' elliptic functions are used. The following goals are achieved within the framework of the method: (a) when representing the algorithms for modelling the dynamics it suffices to use only one chart from the atlas of the phase space manifold, (b) the point in the configuration space of the actual motion lies on the unit sphere, which ensures the best accuracy in numerical procedures, and (c) in the majority of applications the right-hand sides of the equations of perturbed motion depend polynomially on the phase variables, which simplifies the use of computer algebra in analytic theories.