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Symplectic singularities and solvable Hamiltonian mappings

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Języki publikacji
EN
Abstrakty
EN
We study singularities of smooth mappings (…) of R2n into symplectic space (…) by their isotropic liftings to the corresponding symplectic tangent bundle (…). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over (…) endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.
Wydawca
Rocznik
Strony
118--146
Opis fizyczny
Bibliogr. 15 poz.
Twórcy
autor
  • Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakurajousui 3-25-40, Setagaya-Ku, 156-8550 Tokyo, Japan, fukuda@math.chs.nihon-u.ac.jp
autor
  • Institute of Mathematics, Polish Academy of Sciences, Ul. Śniadeckich 8, 00-956 Warsaw, Poland; and Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland, Janeczko@mini.pw.edu.pl
Bibliografia
  • [1] V. I. Arnol’d, S. Guseibn-Zade, A. Varchenko, Singularities of smooth mappings, Russian Math. Surveys 41(6) (1986), 1–21.
  • [2] J. W. Bruce, P. J. Giblin, Curves and Singularities, Cambridge Univ. Press, 1992.
  • [3] P. A. M. Dirac, Generalized Hamiltonian Dynamics, Canad. J. Math. 2 (1950), 129–148.
  • [4] T. Fukuda, Local topological properties of differentiable mappings I, Invent. Math. 65 (1981), 227–250.
  • [5] T. Fukuda, S. Janeczko, Singularities of implicit differential systems and their integrability, Banach Center Publ. 65 (2004), 23–47.
  • [6] T. Fukuda, S. Janeczko, Global properties of integrable implicit Hamiltonian systems, Proc. of the 2005 Marseille Singularity School and Conference, World Scientific (2007), 593–611.
  • [7] T. Fukuda, S. Janeczko, On singularities of Hamiltonian mappings, Geometry and Topology of Caustics-CAUSTICS’06, Banach Center Publ. 82 (2008), 111–124.
  • [8] H. Hofer, E. Zender, Symplectic Invariants and Hamiltonian Dynamics, Birkhaüser Advanced Texts, 1994.
  • [9] G. Ishikawa, Symplectic and Lagrange stabilities of open Whitney umbrellas, Invent. Math. 126 (1996), 215–234.
  • [10] G. Ishikawa, Determinacy, transversality and Lagrange stability, Banach Center Publ. 50 (1999), 123–135.
  • [11] S. Janeczko, On implicit Lagrangian differential systems, Ann. Polon. Math. 74 (2000), 133–141.
  • [12] J. Martinet, Singularities of Smooth Functions and Maps, Cambridge Univ. Press, Cambridge, 1982.
  • [13] J. N. Mather, Solutions of generic linear equations, Dyn. Syst. (1972), 185–193.
  • [14] F. Takens, Implicit differential equations: some open problems, Lecture Notes in Math. 535 (1976), 237–253.
  • [15] A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conf. Ser. in Math., 29, AMS Providence, R.I., 1977.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-bb34fda0-895f-46c3-a57a-2d5ed3c8c019
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