Symplectic theory of completely integrable Hamiltonian systems

Pelayo, Álvaro; Ngọc, San

doi:10.1090/S0273-0979-2011-01338-6

Symplectic theory of completely integrable Hamiltonian systems
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by Álvaro Pelayo and San Vũ Ngọc PDF

Bull. Amer. Math. Soc. 48 (2011), 409-455 Request permission

Abstract:

This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic $4$-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat, and Eliasson; we also give a concise survey of this work. As a motivation, we present a collection of remarkable results proved in the early and mid-1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin and Sternberg, and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes with a discussion of a spectral conjecture for quantum integrable systems.

References

Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
K. Ahara and A. Hattori, $4$-dimensional symplectic $S^1$-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), no. 2, 251–298. MR 1127083
A. Yu. Alekseev, On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom. 45 (1997), no. 2, 241–256. MR 1449971
V. I. Arnol′d, A theorem of Liouville concerning integrable problems of dynamics. , Sibirsk. Mat. Ž. 4 (1963), 471–474 (Russian). MR 0147742
E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny: Fractional bidromy in the vibrational spectrum of hocl. Phys. Rev. Letters, 104 (113002): (2010) 1–4.
M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1
Michèle Audin, Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension $4$, Géométrie symplectique et mécanique (La Grande Motte, 1988) Lecture Notes in Math., vol. 1416, Springer, Berlin, 1990, pp. 1–25 (French). MR 1047474, DOI 10.1007/BFb0097462
Michèle Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR 1106194, DOI 10.1007/978-3-0348-7221-8
O. Babelon, L. Cantini, and B. Douçot: A semi-classical study of the Jaynes–Cummings model. (English summary) J. Stat. Mech. Theory Exp. 2009, no. 7, P07011.
Yves Benoist, Actions symplectiques de groupes compacts, Geom. Dedicata 89 (2002), 181–245 (French, with English summary). MR 1890958, DOI 10.1023/A:1014253511289
Alexey V. Bolsinov and Andrey A. Oshemkov, Singularities of integrable Hamiltonian systems, Topological methods in the theory of integrable systems, Camb. Sci. Publ., Cambridge, 2006, pp. 1–67. MR 2454549
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems, Chapman & Hall/CRC, Boca Raton, FL, 2004. Geometry, topology, classification; Translated from the 1999 Russian original. MR 2036760, DOI 10.1201/9780203643426
A. V. Bolsinov, P. Rikhter, and A. T. Fomenko, The method of circular molecules and the topology of the Kovalevskaya top, Mat. Sb. 191 (2000), no. 2, 3–42 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 1-2, 151–188. MR 1751773, DOI 10.1070/SM2000v191n02ABEH000451
B. Branham and H. Hofer: First steps towards a symplectic dynamics. arXiv:1102.3723
Ricardo Castaño Bernard, Symplectic invariants of some families of Lagrangian $T^3$-fibrations, J. Symplectic Geom. 2 (2004), no. 3, 279–308. MR 2131638
R. Castaño-Bernard and D. Matessi, Some piece-wise smooth Lagrangian fibrations, Rend. Semin. Mat. Univ. Politec. Torino 63 (2005), no. 3, 223–253. MR 2201567
Ricardo Castaño Bernard and Diego Matessi, Lagrangian 3-torus fibrations, J. Differential Geom. 81 (2009), no. 3, 483–573. MR 2487600
Anne-Marie Charbonnel, Comportement semi-classique du spectre conjoint d’opérateurs pseudodifférentiels qui commutent, Asymptotic Anal. 1 (1988), no. 3, 227–261 (French, with English summary). MR 962310
M. S. Child, T. Weston, and J. Tennyson: Quantum monodromy in the spectrum of H2O and other systems: new insight into the level structure of quasi-linear molecules. Mol. Phys., 96 (3) (1999) 371–379.
L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations 28 (2003), no. 9-10, 1527–1566. MR 2001172, DOI 10.1081/PDE-120024521
Yves Colin de Verdière, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable, Math. Z. 171 (1980), no. 1, 51–73 (French). MR 566483, DOI 10.1007/BF01215054
Y. Colin de Verdière and B. Parisse: Équilibre instable en régime semi-classique I : Concentration microlocale. Comm. Partial Differential Equations, 19 (9–10) (1994) 1535–1563.
Yves Colin de Verdière and Bernard Parisse, Équilibre instable en régime semi-classique. II. Conditions de Bohr-Sommerfeld, Ann. Inst. H. Poincaré Phys. Théor. 61 (1994), no. 3, 347–367 (French, with English and French summaries). MR 1311072
Y. Colin de Verdière and J. Vey, Le lemme de Morse isochore, Topology 18 (1979), no. 4, 283–293 (French). MR 551010, DOI 10.1016/0040-9383(79)90019-3
Yves Colin De Verdière, Singular Lagrangian manifolds and semiclassical analysis, Duke Math. J. 116 (2003), no. 2, 263–298. MR 1953293, DOI 10.1215/S0012-7094-03-11623-3
Yves Colin de Verdière and San Vũ Ngọc, Singular Bohr-Sommerfeld rules for 2D integrable systems, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 1, 1–55 (English, with English and French summaries). MR 1987976, DOI 10.1016/S0012-9593(03)00002-8
Richard H. Cushman and Larry M. Bates, Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997. MR 1438060, DOI 10.1007/978-3-0348-8891-2
G. Darboux: Sur le problème de Pfaff, Bulletin des Sciences mathéma. et astrono., 2 série, t. VI; I88z, (1882) 1–46.
Rafael de la Llave, A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175–292. MR 1858536, DOI 10.1090/pspum/069/1858536
Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), no. 6, 687–706. MR 596430, DOI 10.1002/cpa.3160330602
Johannes Jisse Duistermaat and Alvaro Pelayo, Symplectic torus actions with coisotropic principal orbits, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2239–2327 (English, with English and French summaries). Festival Yves Colin de Verdière. MR 2394542
Johannes J. Duistermaat and Alvaro Pelayo, Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 695–718. MR 2496353, DOI 10.1017/S0305004108002077
J. J. Duistermaat and Á. Pelayo: Complex structures on four-manifolds with symplectic two-torus actions. Intern. J. of Math. 22 (2011) 449–463.
J. J. Duistermaat and Á. Pelayo: Topology of symplectic torus actions with symplectic orbits. Revista Matemática Complutense 24 (2011) 59–81.
J. J. Duistermaat: Personal communication, December 2009.
J.-P. Dufour and P. Molino, Compactification d’actions de $\textbf {R}^n$ et variables action-angle avec singularités, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 151–167 (French). MR 1104924, DOI 10.1007/978-1-4613-9719-9_{9}
Jean-Paul Dufour, Pierre Molino, and Anne Toulet, Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 10, 949–952 (French, with English and French summaries). MR 1278158
Holger R. Dullin and San Vũ Ngọc, Vanishing twist near focus-focus points, Nonlinearity 17 (2004), no. 5, 1777–1785. MR 2086150, DOI 10.1088/0951-7715/17/5/012
H. R. Dullin and S. Vũ Ngọc, Symplectic invariants near hyperbolic-hyperbolic points, Regul. Chaotic Dyn. 12 (2007), no. 6, 689–716. MR 2373167, DOI 10.1134/S1560354707060111
Y. Eliashberg and L. Polterovich: Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arXiv:1006.2501.
L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4–35. MR 1036125, DOI 10.1007/BF02566590
L. H. Eliasson: Hamiltonian systems with Poisson commuting integrals, PhD thesis, University of Stockholm, 1984.
N. J. Fitch, C. A. Weidner, L. P. Parazzoli, H. R. Dullin, and H. J. Lewandowski: Experimental demonstration of classical hamiltonian monodromy in the 1 : 1 : 2 resonant elastic pendulum. Phys. Rev. Lett., (2009) (034301).
Hermann Flaschka and Tudor Ratiu, A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 6, 787–809. MR 1422991
A. T. Fomenko (ed.), Topological classification of integrable systems, Advances in Soviet Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1991. Translated from the Russian. MR 1141218, DOI 10.1090/advsov/006
Theodore Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. (2) 70 (1959), 1–8. MR 131883, DOI 10.2307/1969889
K. È. Fel′dman, Hirzebruch genera of manifolds supporting a Hamiltonian circle action, Uspekhi Mat. Nauk 56 (2001), no. 5(341), 187–188 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 5, 978–979. MR 1892568, DOI 10.1070/RM2001v056n05ABEH000446
Mauricio D. Garay, A rigidity theorem for Lagrangian deformations, Compos. Math. 141 (2005), no. 6, 1602–1614. MR 2188452, DOI 10.1112/S0010437X05001478
Andrea Giacobbe, Convexity of multi-valued momentum maps, Geom. Dedicata 111 (2005), 1–22. MR 2155173, DOI 10.1007/s10711-004-1620-y
Viktor L. Ginzburg, Some remarks on symplectic actions of compact groups, Math. Z. 210 (1992), no. 4, 625–640. MR 1175727, DOI 10.1007/BF02571819
Mark Gross and Bernd Siebert, Affine manifolds, log structures, and mirror symmetry, Turkish J. Math. 27 (2003), no. 1, 33–60. MR 1975331
Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. MR 2213573
Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), no. 4, 679–780. MR 2669728, DOI 10.1090/S1056-3911-2010-00555-3
M. Gross and B. Siebert: From real affine geometry to complex geometry, arXiv:math/0703822.
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117, DOI 10.1007/BF01398933
V. V. Kalashnikov, Generic integrable Hamiltonian systems on a four-dimensional symplectic manifold, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 2, 49–74 (Russian, with Russian summary); English transl., Izv. Math. 62 (1998), no. 2, 261–285. MR 1623822, DOI 10.1070/im1998v062n02ABEH000173
Yael Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 43–47. MR 1432457
Yael Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672, viii+71. MR 1612833, DOI 10.1090/memo/0672
M. P. Kharlamov, Topologicheskiĭ analiz integriruemykh zadach dinamiki tverdogo tela, Leningrad. Univ., Leningrad, 1988 (Russian). MR 948454
Frances Kirwan, Convexity properties of the moment mapping. III, Invent. Math. 77 (1984), no. 3, 547–552. MR 759257, DOI 10.1007/BF01388838
Bertram Kostant, Orbits, symplectic structures and representation theory, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. p. 71. MR 0213476
Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR 0294568
B. Kostant and Á. Pelayo: Geometric Quantization, Springer, to appear.
Jarek Kędra, Yuli Rudyak, and Aleksy Tralle, Symplectically aspherical manifolds, J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21. MR 2402905, DOI 10.1007/s11784-007-0048-z
K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. MR 187255, DOI 10.2307/2373157
Maxim Kontsevich and Yan Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321–385. MR 2181810, DOI 10.1007/0-8176-4467-9_{9}
C. Laurent-Gengoux, E. Miranda and P. Vanhaecke: Action-angle coordinates for integrable systems on Poisson manifolds. Preprint, arxiv.0805.1679.
L. M. Lerman and Ya. L. Umanskiĭ, Structure of the Poisson action of $\textbf {R}^2$ on a four-dimensional symplectic manifold. I, Selecta Math. Soviet. 6 (1987), no. 4, 365–396. Selected translations. MR 925264
L. M. Lerman and Ya. L. Umanskiĭ, Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\textbf {R}^2$ in extended neighborhoods of simple singular points. III. Realizations, Mat. Sb. 186 (1995), no. 10, 89–102 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 10, 1477–1491. MR 1361596, DOI 10.1070/SM1995v186n10ABEH000080
Naichung Conan Leung and Margaret Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom. 8 (2010), no. 2, 143–187. MR 2670163
Y. Lin and A. Pelayo: Non-Kähler symplectic manifolds with toric symmetries. Quart. J. Math. 62 (2011) 103–114.
J. Liouville: Note sur l’intégration des équations différentielles de la Dynamique. J. Math. Pures Appl., 20 137–138 (1855). Présentée en 1853.
Gregory Lupton and John Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261–288. MR 1282893, DOI 10.1090/S0002-9947-1995-1282893-4
H. Mineur: Sur les systèmes mécaniques dans lesquels figurent des paramètres fonctions du temps. etude des systemes admettant n intégrales premies uniformes en involution. Extension à ces systèmes des conditions de quantification de Bohr-Sommerfeld. J. Ecole Polytechn., III (1937) (Cahier 1, Fasc. 2 et 3):173–191, 237–270.
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682, DOI 10.1007/978-1-4612-2682-6
Eva Miranda and Nguyen Tien Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 819–839 (English, with English and French summaries). MR 2119240, DOI 10.1016/j.ansens.2004.10.001
E. Miranda and S. Vũ Ngọc: A singular Poincaré lemma. Intern. Math. Res. Not. IMRN 2005 1 27–45.
Dusa McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149–160. MR 1029424, DOI 10.1016/0393-0440(88)90001-0
Nikolaí N. Nekhoroshev, Dmitrií A. Sadovskií, and Boris I. Zhilinskií, Fractional Hamiltonian monodromy, Ann. Henri Poincaré 7 (2006), no. 6, 1099–1211. MR 2267061, DOI 10.1007/s00023-006-0278-4
Juan-Pablo Ortega and Tudor S. Ratiu, A symplectic slice theorem, Lett. Math. Phys. 59 (2002), no. 1, 81–93. MR 1894237, DOI 10.1023/A:1014407427842
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2021152, DOI 10.1007/978-1-4757-3811-7
Alvaro Pelayo, Symplectic actions of 2-tori on 4-manifolds, Mem. Amer. Math. Soc. 204 (2010), no. 959, viii+81. MR 2640344, DOI 10.1090/S0065-9266-09-00584-5
Á. Pelayo and T. S. Ratiu: Applying Hodge theory to dectect Hamiltonian flows. Preprint. arXiv.1005.2163.
Á. Pelayo, T. S. Ratiu and S. Vũ Ngọc: Singular Lagrangian fibrations of Hamiltonian systems. Preprint.
Á. Pelayo and S. Tolman: Fixed points of symplectic periodic flows. Erg. Theory and Dyn. Syst. 31 (2011), in press.
Alvaro Pelayo and San Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math. 177 (2009), no. 3, 571–597. MR 2534101, DOI 10.1007/s00222-009-0190-x
Á. Pelayo and S. Vũ Ngọc: Constructing integrable systems of semitoric type. Acta Math. 206 (2011) 93–125.
Á. Pelayo and S. Vũ Ngọc: Symplectic and spectral theory for spin-oscillators. Preprint. arXiv.1005.439.
Nicolas Roy, The geometry of nondegeneracy conditions in completely integrable systems, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 705–719 (English, with English and French summaries). MR 2188589
Helmut Rüssmann, Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 154 (1964), 285–300 (German). MR 179409, DOI 10.1007/BF01362565
D. A. Sadovskií and B. Zhilinskií: Counting levels within vibrational polyads. J. Chem. Phys., 103 (24) (1995).
D. A. Sadovskií and B. I. Zĥilinskií, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256 (1999), no. 4, 235–244. MR 1689376, DOI 10.1016/S0375-9601(99)00229-7
Christian Sevenheck and Duco van Straten, Rigid and complete intersection Lagrangian singularities, Manuscripta Math. 114 (2004), no. 2, 197–209. MR 2067793, DOI 10.1007/s00229-004-0456-y
J.-M. Souriau, Quantification géométrique, Comm. Math. Phys. 1 (1966), 374–398 (French, with English summary). MR 207332
J.-M. Souriau, Structure of dynamical systems, Progress in Mathematics, vol. 149, Birkhäuser Boston, Inc., Boston, MA, 1997. A symplectic view of physics; Translated from the French by C. H. Cushman-de Vries; Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. MR 1461545, DOI 10.1007/978-1-4612-0281-3
P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699–713. MR 362395, DOI 10.1112/plms/s3-29.4.699
I. Stewart: Quantizing the classical cat. Nature, 430 (2004), 731–732.
Margaret Symington, Generalized symplectic rational blowdowns, Algebr. Geom. Topol. 1 (2001), 503–518. MR 1852770, DOI 10.2140/agt.2001.1.503
Margaret Symington, Four dimensions from two in symplectic topology, Topology and geometry of manifolds (Athens, GA, 2001) Proc. Sympos. Pure Math., vol. 71, Amer. Math. Soc., Providence, RI, 2003, pp. 153–208. MR 2024634, DOI 10.1090/pspum/071/2024634
J. R. Taylor: Classical Mechanics, University Science Books (2005).
Susan Tolman and Jonathan Weitsman, On semifree symplectic circle actions with isolated fixed points, Topology 39 (2000), no. 2, 299–309. MR 1722020, DOI 10.1016/S0040-9383(99)00011-7
John A. Toth, On the quantum expected values of integrable metric forms, J. Differential Geom. 52 (1999), no. 2, 327–374. MR 1758299
John A. Toth and Steve Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111 (2002), no. 1, 97–132. MR 1876442, DOI 10.1215/S0012-7094-02-11113-2
J. Vey, Sur certains systèmes dynamiques séparables, Amer. J. Math. 100 (1978), no. 3, 591–614 (French). MR 501141, DOI 10.2307/2373841
San Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math. 53 (2000), no. 2, 143–217. MR 1721373, DOI 10.1002/(SICI)1097-0312(200002)53:2<143::AID-CPA1>3.0.CO;2-D
San Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology 42 (2003), no. 2, 365–380. MR 1941440, DOI 10.1016/S0040-9383(01)00026-X
San Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math. 208 (2007), no. 2, 909–934. MR 2304341, DOI 10.1016/j.aim.2006.04.004
San Vũ Ngọc, Symplectic techniques for semiclassical completely integrable systems, Topological methods in the theory of integrable systems, Camb. Sci. Publ., Cambridge, 2006, pp. 241–270. MR 2454557
S. Vũ Ngọc: Systèmes Integrables Semi-Classiques : du Local au Global. Pan. et Synthèses SMF, 22, 2006.
S. Vũ Ngọc: Symplectic inverse spectral theory for pseudodifferential operators. To appear in Geometric Aspects of Analysis and Mechanics, in honor of the 65th birthday of Hans Duistermaat; to be published by Birkhäuser, Boston, 2011.
S. Vũ Ngọc and C. Wacheux: Smooth normal forms for integrable hamiltonian systems near a focus-focus singularity. hal preprint oai:hal.archives-ouvertes.fr: hal-00577205, arXiv: 1103.3282.
John Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J. Math. 58 (1936), no. 1, 141–163. MR 1507138, DOI 10.2307/2371062
Nguyen Tien Zung, Kolmogorov condition for integrable systems with focus-focus singularities, Phys. Lett. A 215 (1996), no. 1-2, 40–44. MR 1396244, DOI 10.1016/0375-9601(96)00219-8
Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities, Compositio Math. 101 (1996), no. 2, 179–215. MR 1389366
Nguyen Tien Zung, A note on degenerate corank-one singularities of integrable Hamiltonian systems, Comment. Math. Helv. 75 (2000), no. 2, 271–283. MR 1774706, DOI 10.1007/PL00000375
N. T. Zung, Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems, Regul. Chaotic Dyn. 12 (2007), no. 6, 680–688. MR 2373157, DOI 10.1134/S156035470706010X
Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form, Ann. of Math. (2) 161 (2005), no. 1, 141–156. MR 2150385, DOI 10.4007/annals.2005.161.141
Alan Weinstein, Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Lett. Math. Phys. 56 (2001), no. 1, 17–30. EuroConférence Moshé Flato 2000, Part I (Dijon). MR 1848163, DOI 10.1023/A:1010913023218