A characterization of submanifolds by a homogeneity condition

A very short proof of the following smooth homogeneity theorem of D. Repovs, E. V. Scepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x,y in N there exist their neighborhoods Ux and Uy in M and a diffeomorphism h : Ux \to Uy such that h(x)=y and h (Ux \cap N) = Uy \cap N. Then N is a smooth submanifold of M.Comment: 4 pages, meaning-distorting typos correcte

Bi-Hamiltonian structures and singularities of integrable systems

Hamiltonian system on a Poisson manifold M is called integrable if it possesses sufficiently many commuting first integrals f1, . . . fs which are functionally independent on M almost everywhere. We study the structure of the singular set K where the differentials df1, . . . , dfs become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers

Geometry of integrable dynamical systems on 2-dimensional surfaces

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions, and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.Comment: 31 pages, 12 figures, submitted to a special issue of Acta Mathematica Vietnamic

Бифуркации интегрируемых механических систем с магнитным полем на поверхностях вращения

On a surface homeomorphic to 2-sphere, we study a natural mechanical system with amagnetic field that is invariant under the $S^1$-action. For singular points of rank 0 of themomentum mapping, a criterion for non-degeneracy is obtained, the type of non-degeneratesingular points (center-center and focus-focus) is determined, bifurcations of typical degeneratesingular points are described (integrable Hamiltonian Hopf bifurcation of two types). For familiesof singular circles of rank 1 of the momentum mapping (consisting of relative equilibriums of thesystem) their parametric representation is obtained, nondegeneracy criterion is proved, the typeof nondegenerate (elliptic and hyperbolic) and typical degenerate (parabolic) singular circlesis determined. The parametric representation of the bifurcation diagram of the momentummapping is obtained. Geometric properties of the bifurcation diagram and the bifurcationcomplex are described in the case when the functions defining the system are in general position.The topology of nonsingular isoenergy 3-dimensional manifolds is determined, the topology ofthe Liouville foliation on them is described up to the rough Liouville equivalence (in terms ofFomenko’s atoms and molecules). The “splitting” hyperbolic singularities of rank 1 are described,which are topologically unstable bifurcations of the Liouville foliation.На поверхности, гомеоморфной 2-мерной сфере, изучается натуральная механическая система с магнитным полем, инвариантная относительно $S^1$-действия. Для особых точек ранга 0 отображения момента получен критерий невырожденности, определен тип невырожденных особых точек (центр-центр и фокус-фокус), описаны бифуркации типичных вырожденных особых точек (интегрируемая гамильтонова бифуркация Хопфа двух типов). Для семейств особых окружностей ранга 1 отображения момента (состоящих из относительных положений равновесия системы) получено их параметрическое задание, доказан критерий невырожденности, определен тип невырожденных (эллиптические и гиперболические) и типичных вырожденных (параболические) особых окружностей. Получено параметрическое задание бифуркационной диаграммы отображения момента. Описаны геометрические свойства бифуркационной диаграммы и бифуркационного комплекса в случае, когда задающие систему функции находятся в общем положении. Определена топология неособых изоэнергетических 3-мерных многообразий, описана топология слоения Лиувилля на них с точностью до грубой лиувиллевой эквивалентности (в терминах атомов и молекул Фоменко). Описаны “расщепляющиеся” гиперболические особенности ранга 1, являющиеся топологически неустойчивыми бифуркациями слоения Лиувилля

Topology of energy surfaces and existence of transversal Poincar\'e sections

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.Comment: 10 pages, LaTe

Foliations of Isonergy Surfaces and Singularities of Curves

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level. We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure

Integrable systems with linear periodic integral for the Lie algebra \eLie

Integrable systems with a linear periodic integral for the Lie algebra \eLie are considered. One investigates singulariries of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville tori, topology of isoenergy surfaces and other topological properties of such systems

Topological classification of hamiltonians in some classical cases of integrability of hamiltonian systems

The work is aimed at obtaining complete classification of isoenergetic surfaces for classical cases of integrability with an accuracy up to rough equivalence. A rough topological classification of isoenergetic surfaces for main classical cases of integrability in solid body dynamics is obtained. Specifically, the bottovost of an additional integral, for all studied cases of integrability is provedAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

TRACE ANALYSIS BY LASER-EXCITED ATOMIC FLUORESCENCE WITH ATOMIZATION IN A PULSED PLASMA

The possibilities of plasma atomization for laser fluorescence trace analysis are discussed. Pulsed hot hollow cathode discharge was used for analysis of solutions and powdered samples. The high voltage spark and laser-induced breakdown (laser spark) were used as atomizers of metal-containing atmospheric aerosols. Detection limits were improved by means of temporal background selection