We discuss the Euler approach to construction and to investigation of the superintegrable systems with additional quadratic and cubic integrals of motions.
We consider two well-known integrable systems on the plane using the concept of natural Poisson bivectors on Riemaninan manifolds. Geometric approach to construction of variables of separation and separated relations for the generalized Henon-Heiles system and the generalized system with quartic potential is discussed in detail.
The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which is against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, both the initial and perturbed Kepler systems are isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.
ДЛЯ НЕКОТОРЫХ СИСТЕМ С ИНТЕГРАЛОМ ДВИЖЕНИЯ ЧЕТВЕРТОЙ СТЕПЕНИПостроены переменные разделения для интегрируемых деформаций Яхьи в случае волчка Ковалевской и системы Чаплыгина на сфере. В общем случае соответствующие квадратуры представляют собой отображение Абеля-Якоби на двумерном подмногообразии якобиана алгебраической кривой рода три, ко-торая не является гиперэллиптической.Ключевые слова: бигамильтонова геометрия, разделение переменных, волчок Кова-левской.
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