Recent zbMATH articles in MSC 70E40https://zbmath.org/atom/cc/70E402024-03-13T18:33:02.981707ZWerkzeugTopological analysis of pseudo-Euclidean Euler top for special values of the parametershttps://zbmath.org/1528.370492024-03-13T18:33:02.981707Z"Altuev, Murat K."https://zbmath.org/authors/?q=ai:altuev.murat-k"Kibkalo, Vladislav A."https://zbmath.org/authors/?q=ai:kibkalo.vladislav-aleksandrovichSummary: An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations.Gyrostatic Suslov problemhttps://zbmath.org/1528.700042024-03-13T18:33:02.981707Z"Maciejewski, Andrzej J."https://zbmath.org/authors/?q=ai:maciejewski.andrzej-j"Przybylska, Maria"https://zbmath.org/authors/?q=ai:przybylska.mariaSummary: In this paper, we investigate the gyrostat under influence of an external potential force with the Suslov nonholonomic constraint: the projection of the total angular velocity onto a vector fixed in the body vanishes. We investigate cases of free gyrostat, the heavy gyrostat in the constant gravity field, and we discuss certain properties for general potential forces. In all these cases, the system has two first integrals: the energy and the geometric first integral. For its integrability, either two additional first integrals or one additional first integral and an invariant \(n\)-form are necessary. For the free gyrostat we identify three cases integrable in the Jacobi sense. In the case of heavy gyrostat three cases with one additional first integral are identified. Among them, one case is integrable and the non-integrability of the remaining cases is proved by means of the differential Galois methods. Moreover, for a distinguished case of the heavy gyrostat a co-dimension one invariant subspace is identified. It was shown that the system restricted to this subspace is super-integrable, and solvable in elliptic functions. For the gyrostat in general potential force field conditions of the existence of an invariant \(n\)-form defined by a special form of the Jacobi last multiplier are derived. The class of potentials satisfying them is identified, and then the system restricted to the corresponding invariant subspace of co-dimension one appears to be integrable in the Jacobi sense.Stability of stationary motions of mechanical systems with a particular Goryachev-Chaplygin integralhttps://zbmath.org/1528.700072024-03-13T18:33:02.981707Z"Novikov, M. A."https://zbmath.org/authors/?q=ai:novikov.m-a(no abstract)