Abstract
We study singularities of smooth mappings F̄ of ℝ2n into symplectic space (ℝ2n , ω̇) by their isotropic liftings to the corresponding symplectic tangent bundle (Tℝ2n,w). 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 F̄ 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.
References
[1] V. I. Arnol’d, S. Guseibn-Zade, A. Varchenko, Singularities of smooth mappings, Russian Math. Surveys 41(6) (1986), 1-21.Search in Google Scholar
[2] J. W. Bruce, P. J. Giblin, Curves and Singularities, Cambridge Univ. Press, 1992.10.1017/CBO9781139172615Search in Google Scholar
[3] P. A. M. Dirac, Generalized Hamiltonian Dynamics, Canad. J. Math. 2 (1950), 129-148.10.4153/CJM-1950-012-1Search in Google Scholar
[4] T. Fukuda, Local topological properties of differentiable mappings I, Invent. Math. 65 (1981), 227-250.10.1007/BF01389013Search in Google Scholar
[5] T. Fukuda, S. Janeczko, Singularities of implicit differential systems and their integrability, Banach Center Publ. 65 (2004), 23-47.10.4064/bc65-0-2Search in Google Scholar
[6] T. Fukuda, S. Janeczko, Global properties of integrable implicit Hamiltonian systems, Proc. of the 2005 Marseille Singularity School and Conference, World Scientific (2007), 593-611.10.1142/9789812707499_0024Search in Google Scholar
[7] T. Fukuda, S. Janeczko, On singularities of Hamiltonian mappings, Geometry and Topology of Caustics-CAUSTICS’06, Banach Center Publ. 82 (2008), 111-124.Search in Google Scholar
[8] H. Hofer, E. Zender, Symplectic Invariants and Hamiltonian Dynamics, Birkhaüser Advanced Texts, 1994.10.1007/978-3-0348-8540-9Search in Google Scholar
[9] G. Ishikawa, Symplectic and Lagrange stabilities of open Whitney umbrellas, Invent. Math. 126 (1996), 215-234.10.1007/s002220050095Search in Google Scholar
[10] G. Ishikawa, Determinacy, transversality and Lagrange stability, Banach Center Publ. 50 (1999), 123-135.10.4064/-50-1-123-135Search in Google Scholar
[11] S. Janeczko, On implicit Lagrangian differential systems, Ann. Polon. Math. 74 (2000), 133-141.10.4064/ap-74-1-133-141Search in Google Scholar
[12] J. Martinet, Singularities of Smooth Functions and Maps, Cambridge Univ. Press, Cambridge, 1982.Search in Google Scholar
[13] J. N. Mather, Solutions of generic linear equations, Dyn. Syst. (1972), 185-193.10.1016/B978-0-12-550350-1.50020-5Search in Google Scholar
[14] F. Takens, Implicit differential equations: some open problems, Lecture Notes in Math. 535 (1976), 237-253.10.1007/BFb0080503Search in Google Scholar
[15] A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conf. Ser. in Math., 29, AMS Providence, R.I., 1977. 10.1090/cbms/029Search in Google Scholar
© by Stanislaw Janeczko
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