
Scientific paper ID 1642 : 2018/3
COMPUTING OF NONINTEGRABILITY WITH COMPUTER ALGEBRA METHODS
Georgi Georgiev Abstract. In this paper it is shown that the Hamiltonian system with Dyson potential is analytical nonintegrable and formal non integrable. The Dyson system, appeared in 1962 by Dyson statistical physics research of the energy levels of onedimensional Coulomb’s gas. There is connection between the of system of point vortices  this is result by Calogero and Perelomov in 1978. Hamiltonian system has two integrals, that is why for integrability it is important the first nontrivial case . Borisov and Kozlov in 1998 had proved that the system in case is nonintegrable in analytical first integrals using splitting separarises method. In this paper it is proved the same statement, but way is different. Proof based of Duistermaat’s idea when a system has a family of periodic solutions around equilibrium, and if the period function is infinitely branched, then if the system has additional analytical first integral  this integral is a constant. We call that the system with Hamiltonian H formally integrable if there exist formal power series in involution, where are functionally independent and Taylor expansion of H is a formal power series in . An asymptotic behavior near equilibrium is like an integrable system. The formal integrability gives asymptotic information about the flow. For the proof of formal nonintegrability in the paper is used Theory of Ziglin  MoralesRuizRamis. Computer algebra method is the Kovacic algorithm using a software for education, engineering and research Maple.
Потенциал на Дайсън Хамилтонови системи Неинтегруемост Компютърни алгебри.Dyson potential Hamiltonian systems Nonintegrability Computer Algebra Georgi Georgiev BIBLIOGRAPHY [1] Dyson F.J., (1962), Statistical theory of the enrgy levels of complex systems, I, II, III, J. Math Phys., 3, 140156, 157165, 166175. [2] Calogero F., Perelomov A. M., (1978), Properties of Certain Matrices Related to the Equilibrium Configuration of the OneDimensional ManyBody Problems with the Pair Potentials, and , Commun. math. Phys. 59, no.12, 109116. [3] Kozlov V.V., (1995), Symmetries, Topology, and Resonace in Hamiltonian Mechanics, Izhevsk, Udmurt Univ. Moskow (in Russian). [4] Borisov A. V., Kozlov V.V., (1998), Nonitegrability Of a System of Interacting Particles with the Dyson Potential, Doklady Akademii Nauk, Vol 306, no. 1, 30 31. [5] Duistermaat J. J., (1984), Nonintegrability of the 1:1:2 resonance, Ergod. Th. & Dynam. Sys., 4, 553 568. [6] Christov O., Georgiev G. (2015), On the integrability of a system describing the stationary solutions in BoseFermi mixtures, Chaos Solitons and Fractals, 77, 8, 138 148. [7] MoralesRuiz J., (1999), Differential Galois Theory and Nonintegrability of Hamiltonian Systems, Birkheauser. [8] Yoshida H., (1988), NonIntegrability of the Truncated Toda Lattice Hamiltonian at Any Order, Commun. Math. Phys., 116, 529 538. 