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Billiards of Variable Configuration and Billiards with Slip** in Hamiltonian Geometry and Topology
A class of billiards is found, the geometry of which can change with a change in the energy of a ball moving on a ‘‘billiard table.’’ Such billiards are called force or evolutionary billiards. They make it pos...
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Article
Billiards with Changing Geometry and Their Connection with the Implementation of the Zhukovsky and Kovalevskaya Cases
The paper presents a class of billiards with varying geometry, the so-called force or evolutionary billiards, which enable us to realize, in the sense of Liouville equivalence, the well-known cases of Zhukovs...
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Article
Liouville Foliations of Topological Billiards with Slip**
In the paper, a new class of integrable billiards, namely, billiards with slip**, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but ...
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Article
Noncompact Bifurcations of Integrable Dynamic Systems
In the theory of integrable Hamiltonian systems, an important role is played by the study of Liouville foliations and bifurcations of their leaves. In the compact case, the problem is solved, but the noncompac...
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Article
Implementation of Integrable Systems by Topological, Geodesic Billiards with Potential and Magnetic Field
In the paper, eight classes of integrable billiards are studied; in particular, classes introduced by the authors: elementary, topological, billiard books, billiards on the Minkowski plane, geodesic billiards ...
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Article
Billiards and Integrability in Geometry and Physics. New Scope and New Potential
Description of bifurcations and symmetries of integrable systems is an important branch of geometry that has many applications. Important results have been obtained recently in the descriptions of bifurcations...
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Article
Modeling Nondegenerate Bifurcations of Closures of Solutions for Integrable Systems with Two Degrees of Freedom by Integrable Topological Billiards
It is well known that surgeries of closures of solutions for integrable nondegenerate Hamiltonian systems with two degrees of freedom at a level of constant energy are classified by the so-called 3-atoms. Thes...
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Article
Integrable billiards model important integrable cases of rigid body dynamics
Abstract—A generalized billiard is considered, in which a point moves on a locally flat surface obtained by isometrically gluing together several plane domains along boundaries being arcs of confocal quadrics....
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Chapter
Topological Classification of Geodesic Flows on Revolution 2-Surfaces with Potential
The paper is devoted to a short explanation of the topological classification (up to Liouville equivalence) of the integrable geodesic flows of two-dimensional surfaces of revolution with potential. The classi...
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Article
Geometry, dynamics and different types of orbits
This work provides an outline of several results concerning topology, Lie algebra, orbits and dynamics of some integrable systems on them. All the results in this paper were obtained by the authors and the par...
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Article
Each finite group is a symmetry group of some map (an “Atom”-bifurcation)
Maps are studied, i.e., cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms which encode bifurcations of Liouville fibrations of non-degenerate integrable Hamiltonian systems. Any ...
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Article
Symmetries groups of nice Morse functions on surfaces
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Article
Computer modeling of curves and surfaces
The first part of this work deals with the investigation and modeling of foliations generated by dynamic systems on their phase spaces and configuration spaces. In the second part, we speak in greater detail a...
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Article
Viktor Antonovich Sadovnichii. A tribute in honor of his seventieth birthday
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Article
Riemannian Geometry
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Article
Exact topological classification of Hamiltonian flows on smooth two-dimensional surfaces
The present paper contains an exact topological classification of all nondegenerate Hamiltonian systems on smooth closed two-dimensional surfaces. Bibliography: 8 titles.
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Article
Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body
Physical and mechanical systems with four-dimensional phase space are considered. The classification of nondegenerate integral systems is studied. A “physical zone,’ i.e., the systems connected with real physi...
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Chapter
Orbital Isomorphism Between Two Classical Integrable Systems
We describe the orbital invariants of two famous integrable systems (the Euler case in rigid body dynamics and the Jacobi problem) and show that these systems are orbitally topologically equivalent.
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Book
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Chapter
Connected Sums
Above, we have already episodically used the connected sum operation on manifolds. Now we shall investigate it in more detail for the three-dimensional case.
