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SINGULAR COTANGENT MODEL

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  18 December 2014

CARLOS CURRÁS-BOSCH
CARLOS CURRÁS-BOSCH*
Affiliation:
Departament d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain e-mail: [email protected]
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Abstract

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Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent bundle of the desingularized level is used to perform the construction, and the c.i.s. obtained looks like the simplest one associated to the affine structure. This method of construction is used to provide several examples of c.i.s. with different kinds of non-degenerate singularities.

MSC classification

Primary: 53D05: Symplectic manifolds, general
Secondary: 37J35: Completely integrable systems, topological structure of phase space, integration methods 37G05: Normal forms
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 415 - 430
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Arnold, V. I., Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60 (Springer-Verlag, New York, 1989).Google Scholar
2.Audin, M., Cannas da Siva, A. and Lerman, E., Symplectic geometry of integrable systems, Advanced Courses in Mathematics (CRM Barcelona, Birkhäuser, 2003).Google Scholar
3.Bolsinov, A. V. and Fomenko, A. T., Integrable Hamiltonian systems, geometry, topological classification (Chapman & Hall/CRC, 2004).Google Scholar
4.Boucetta, M. and Molino, P., Géométrie globale des systèmes Hamiltoniennes complétement intégrables, C. R. Acad. Sci. Paris, I 308 (13) (1989), 421–424.Google Scholar
5.Currás-Bosch, C., Decomposition of integrable and non-degenerate Hamiltonian systems with non-zero ellipticity degree, preprint.Google Scholar
6.Currás-Bosch, C. and Molino, P., Holonomie, suspensions et classifications pour les feuilletages Lagrangiens C. R. Acad. Sci. Paris, I 326 11 (1989), 1317–1320.CrossRefGoogle Scholar
7.Delzant, T., Hamiltoniens periodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (3) (1988), 315–339.Google Scholar
8.Duistermaat, J. J., On global action-angle coordinates, Commun. Pure Appl. Math. 33 (6) (1980), 687–706.Google Scholar
9.Eliasson, L. H., Normal forms for Hamiltonian systems with Poisson commuting integrals, PhD Thesis (University of Stockholm, 1984).Google Scholar
10.Fomenko, A. T., Topological classification of integrable systems, in Advances in Soviet Mathematics, vol. 6 (AMS, Providence R. I., 1991).Google Scholar
11.Lerman, E. and Umanskii, Y., Classification of 4-dimensional integrable Hamiltonian systems, in Methods of qualitative theory of bifurcations (Iz.Garkov University, Gorki, 1988), 67–76.Google Scholar
12.Mineur, H., Réduction des systèmes mécaniques à n degrès de liberté, J. Math. Pure Appl. IX, 15 (1936), 385–389.Google Scholar
13.Miranda, E., On symplectic linearization of singular Lagrangian foliations (Tesi University de Barcelona, 2003).Google Scholar
14.Miranda, E. and San, V. N., A singular Poincaré Lemma, Int. Math. Res. Notices (2005), 27–45.CrossRefGoogle Scholar
15.Miranda, E. and Zung, N. T., Equivariant normal forms for non-degenerate singular orbits of integrable Hamiltonian systems Ann. Sci. Ec. Norm. Sup., 37 (2004), 819–839.Google Scholar
16.Molino, P., Action-angle with singularities and non-degenerate integrable systems, (preprint).Google Scholar
17.Toulet, A., Classification des systèmes intégrables en dimension 2 (Thèse University Montpellier II, 1996).Google Scholar
18.San, V. N., On semi-global invariants for focus-focus singularities Topology 42 (2003), 365–380.Google Scholar
19.Zung, N. T., Symplectic topology of integrable Hamiltonian systems PhD Thesis (University of Strasbourg, 1994).Google Scholar
20.Zung, N. T., Symplectic topology of integrable Hamiltonian systems I Compos. Math. 101 (1996), 179–219.Google Scholar
21.Zung, N. T., Symplectic topology of integrable Hamiltonian systems II Compos. Math. 138 (2003), 125–156.Google Scholar
22.Weinstein, A., Lectures on symplectic manifolds, Regional Conference Series in Mathematics (American Mathematical Society, Providence, RI, 1976).Google Scholar
23.Williamson, J., On the algebraic problem concerning the normal form of linear dynamical systems, Am. J. Math. 58 (1) (1936), 141–163.Google Scholar