Topological Atlas of the Kovalevskaya Top in a Double Field

Kharlamov, Mikhail P.; Ryabov, Pavel E.

doi:10.1007/s10958-017-3387-3

Cited by 7 publications

(1 citation statement)

References 42 publications

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“…There exists a series of works devoted to the topological analysis of system (1) (see, e.g., [Ryabov P.E. et al, 2012], [Kharlamov and Ryabov, 2017] and the bibliography therein). The analysis was made on the basis of the methods and approaches described in [Bolsinov and Fomenko, 1999].…”

Section: Introductionmentioning

confidence: 99%

On motions of a conservative system on invariant manifolds

Irtegov

Titorenko

2018

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In [Irtegov and Burlakova, 2017], the algorithms for the qualitative analysis of conservative systems have been presented. These are based on the Routh-Lyapunov method [Lyapunov, 1954] and some its modifications [Irtegov and Titorenko, 2009] as well as computer algebra methods [Cox, Little, and O’Shea, 1997]. In the paper the application of the algorithms is demonstrated by analysing a conservative system, the study of which is also of interest. We conduct qualitative analysis for the differential equations describing the rotational motion of a rigid body with a fixed point in two constant force fields. Similar problems arise, e.g., in space dynamics [Sarychev and Gutnik, 2015], quantum mechanics [Adler, Marikhin, and Shabat, 2012], [Smirnov, 2008]. In the phase space of the problem, we isolate the invariant manifolds of maximal dimension and study the equations of motion on them. For these equations, solutions (and their families) corresponding in the original phase space of the problem to permanent rotations and pendulum-like oscillations of the body as well as the invariant manifolds of 2nd level, which these solutions belong to, have been found and their Lyapunov’s stability has been investigated. The possibility of stabilization for the motions of conservative systems, whose stability conditions have the form of some constraints on the constants of first integrals, is discussed.

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Section: Introductionmentioning

confidence: 99%

On motions of a conservative system on invariant manifolds

Irtegov

Titorenko

2018

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Regular precession of a rigid body in two uniform fields

Ol’shanskii¹

2023

Mechanics Research Communications

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Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Belozerov¹

2022

Sb. Math.

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We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

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Topological Atlas of the Kovalevskaya Top in a Double Field

Cited by 7 publications

References 42 publications

On motions of a conservative system on invariant manifolds

On motions of a conservative system on invariant manifolds

Regular precession of a rigid body in two uniform fields

Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

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