New cases of integrability in the spatial dynamics of a rigid body

Abstract

The results of this work appeared due to investigation of a certain problem on the motion of a rigid bodyin a resistant medium [1, 2], where it was necessary todeal with first integrals of dynamic systems with nonstandard properties. Specifically, the integrals wereneither analytical nor smooth, and for certain sets,they were even discontinuous. The last circumstancesallowed us to completely analyze all phase trajectoriesand to indicate their properties, which possessed“roughness” and were retained for the systems of amore general form with certain nontrivial encapsulatedtype symmetries. Therefore, it is of interest toinvestigate sufficiently wider classes of systems withsimilar properties, specifically, those taken from thedynamics of a rigid body interacting with a medium.New cases of integrability in the problem of spatialmotion of the rigid body in the resistant medium willbe presented.SYSTEMS WITH SYMMETRIES AND VARIABLE DISSIPATIONWITH A ZERO MEANWe studied the systems of ordinary differentialequations having at least one periodic phase coordinate. The systems under study possess symmetry properties such that their phase volume is retained on average for the period by the periodic coordinate. Forexample, the pendulum system with a smooth andperiodic righthand partα