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

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

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

TIME-RESOLVED LASER FLUORESCENT ANALYSIS OF SOLUTIONS, AEROSOL-CONTAINING AND SOLID SAMPLES

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