Monodromy of the fibre with oscillatory singular point of type $1:(-2)$

N. N. Nekhoroshev

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 3, pp. 413-541.

Résumé

In the present work, we prove the existence of fractional monodromy in a large class of compact Lagrangian fibrations of four-dimensional symplectic manifolds. These fibrations are considered in the neighbourhood of the singular fibre $\lambda_0$, that has a single singular point corresponding to a nonlinear oscillator with frequencies in $1:(-2)$ resonance. We compute the matrices of monodromy defined by going around the fibre $\lambda_0$. For all fibrations in the class and for an appropriate choice of the basis in the one-dimensional homology group of the torus, these matrices are the same. The elements of the monodromy matrix are rational and there is a non-integer element among them. This work is a continuation of the analysis in [20, 21, 39] where the matrix of fractional monodromy was computed for most simple particular fibrations of the class.

Bibliographie

[1] Atiyah M. F., “Convexity and commuting Hamiltonians”, Bull. London Math. Soc., 14:1 (1982), 1–15 | DOI | MR | Zbl

[2] Bambusi D., Nekhoroshev N. N., “A property of exponential stability in nonlinear wave equations near the fundamental linear mode”, Phys. D, 122:1–4 (1998), 73–104 | DOI | MR | Zbl

[3] Colin de Verdière Y., Vũ Ngȯc S., “Singular Bohr – Sommerfeld rules for 2D integrable systems”, Ann. Sci. École Norm. Sup. (4), 36:1 (2003), 1–55 | MR | Zbl

[4] Cushman R., “Geometry of the energy momentum mapping of the spherical pendulum”, CWI Newslett., 1983, no. 1, 4–18 | MR

[5] Cushman R. H., Bates L. M., Global aspects of classical integrable systems, Birkhäuser, Basel, 1997, 435 pp. | MR | Zbl

[6] Cushman R., Duistermaat J. J., “Non-Hamiltonian monodromy”, J. Differential Equations, 172:1 (2001), 42–58 | DOI | MR | Zbl

[7] Cushman R. H., Duistermaat J. J., “The quantum mechanical spherical pendulum”, Bull. Amer. Math. Soc. (N. S.), 19:2 (1988), 475–479 | DOI | MR | Zbl

[8] Cushman R. H., Vũ Ngȯc S., “Sign of the monodromy for Liouville integrable systems”, Ann. Henri Poincaré, 3:5 (2002), 883–894 | DOI | Zbl

[9] Dhont G., Sadovskii D. A., Zhilinskii B. I., Boudon V., “Analysis of the «unusual» vibrational components of triply degenerate vibrational mode $\nu_6$ of Mo(CO)$_6$ based on the classical interpretation of the effective rotation–vibration Hamiltonian”, J. Molecular Spectroscopy, 201 (2000), 95–108 | DOI

[10] Duistermaat J. J., “On global action-angle coordinates”, Comm. Pure Appl. Math., 33:6 (1980), 687–706 | DOI | MR | Zbl

[11] Dullin H., Giacobbe A., Cushman R., “Monodromy in the resonant swing spring”, Phys. D, 190:1–2 (2004), 15–37 | DOI | MR | Zbl

[12] Efstathiou K., Cushman R. H., Sadovskii D. A., “Fractional monodromy in the $1:(-2)$ resonance”, Adv. Math., 209:1 (2007), 241–273 | DOI | MR | Zbl

[13] Faure F., Zhilinskii B. I., “Topological Chern indices in molecular spectra”, Phys. Rev. Lett., 85:5 (2000), 960–963 | DOI

[14] Giacobbe A., Cushman R. H., Sadovskii D. A., Zhilinskii B. I., “Monodromy of the quantum $1:1:2$ resonant swing spring”, J. Math. Phys., 45:12 (2004), 5076–5100 | DOI | MR | Zbl

[15] Kosin I. N., Roberts R. M., “Monodromy in the spectrum of a rigid symmetric top molecule in an electric fields”, J. Chem. Phys., 118:23 (2003), 10523–10533 | DOI

[16] Kudryavtseva E. A., “Generalization of geometric Poincaré theorem for small perturbations”, Regul. Chaotic Dyn., 3:2 (1998), 46–66 | DOI | MR | Zbl

[17] Lerman L. M., Umanskiy Ya. L., Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects), Transl. Math. Monogr., 176, AMS, Providence, R.I., 1998, 177 pp. | MR | Zbl

[18] Michel L., Zhilinskii B. I., “Symmetry, invariants, topology: Basic tools”, Phys. Rep., 341:1–6 (2001), 11–84 | DOI | MR | Zbl

[19] Nekhoroshev N. N., “Generalizations of Gordon theorem”, Regul. Chaotic Dyn., 7:3 (2002), 239–247 | DOI | MR | Zbl

[20] Nekhoroshev N. N., Sadovskii D. A., Zhilinskii B. I., “Fractional monodromy of resonant classical and quantum oscillators”, C. R. Math. Acad. Sci. Paris, 335:11 (2002), 985–988 | DOI | MR | Zbl

[21] Nekhoroshev N. N., Sadovskii D. A., Zhilinskii B. I., “Fractional Hamiltonian monodromy”, Ann. Henri Poincaré, 7:6 (2006), 1099–1211 | DOI | MR | Zbl

[22] Nguyen T. Z., “A note on focus-focus singularities”, Differential Geom. Appl., 7:2 (1997), 123–130 | DOI | MR | Zbl

[23] Nguyen T. Z., “Another note on focus-focus singularities”, Lett. Math. Phys., 60:1 (2002), 87–99 | DOI | MR | Zbl

[24] Sadovskii D. A., Zhilinskii B. I., “Qualitative analysis of vibration–rotation Hamiltonians for spherical top molecules”, Molecular Phys., 65:1 (1988), 109–128 | DOI

[25] Sadovskii D. A., Zhilinskii B. I., “Monodromy, diabolic points, and angular momentum coupling”, Phys. Lett. A, 256:4 (1999), 235–244 | DOI | MR | Zbl

[26] Vũ Ngȯc S., “Quantum monodromy in integrable systems”, Comm. Math. Phys., 203:2 (1999), 465–479 | DOI | MR | Zbl

[27] Zhilinskii B. I., “Symmetry, invariants, and topology in molecular models: Symmetry, invariants, topology”, Phys. Rep., 341:1–6 (2001), 85–171 | DOI | MR | Zbl

[28] Arnol'd V. I., Mathematical methods of classical mechanics, Grad. Texts in Math., 60, 2nd ed., Springer, New York, 1989 | DOI | MR | MR

[29] Arnold V. I., Gusein-Zade S. M., Varchenko A. N., Singularities of differentiable maps: In 2 vols., Monogr. Math., 8283, Birkhäuser, Basel, 1985, 1988 | MR

[30] Arnol'd V. I., Givental' A. B., “Symplectic geometry”, Dynamical systems IV: Symplectic geometry and its applications, Encyclopaedia Math. Sci., 4, eds. V. I. Arnol'd, S. P. Novikov, Springer, Berlin, 2001, 1–138 | MR

[31] Arnol'd V. I., Kozlov V. V., Neïshtadt A. I., Mathematical aspects of classical and celestial mechanics, Encyclopaedia Math. Sci., 3, Springer, Berlin, 2006 | MR | MR | Zbl

[32] Bishop R. L., Crittenden R. J., Geometry of manifolds, AMS, Chelsea, R.I., 2001, 273 pp. | MR | MR | Zbl

[33] Borisov A. V., Mamaev I. S., Poisson structures and Lie algebras in Hamiltonian mechanics, R Dynamics, Izhevsk, 1999 (Russian) | MR

[34] Guillemin V., Sternberg Sh., Geometric asymptotics, Math. Surveys Monogr., 14, AMS, Providence, R.I., 1977, xviii+474 pp. | DOI | MR | Zbl

[35] Karasev N. V., Maslov V. P., Nonlinear Poisson brackets, geometry and quantization, Transl. of Math. Monogr., 119, AMS, Providence, R.I., 1993, xi + 366 pp. | MR | MR | Zbl

[36] Kozlov V. V., Symmetries, topology and resonances in Hamiltonian mechanics, Ergeb. Math. Grenzgeb. (3), 31, Springer, Berlin, 1996, xii+378 pp. | MR | MR

[37] Matveev V. S., “Integrable Hamiltonian systems with two degrees of freedom. Topological structure of saturated neighborhoods of points of focus–focus and saddle–saddle types”, Sb. Math., 187:4 (1996), 495–524 | DOI | DOI | MR | Zbl

[38] Nekhoroshev N. N., “Two theorems on the action–angle variables”, Uspekhi Mat. Nauk, 24:5(149) (1969), 237–238 (Russian) | MR | Zbl

[39] Nekhoroshev N. N., “Fractional monodromy in the case of arbitrary resonances”, Sb. Math., 198:3–4 (2007), 383–424 | DOI | DOI | MR | Zbl

[40] Nekhoroshev N. N., “Action-angle variables and their generalization”, Trans. Moscow Math. Soc., 26 (1972), 180–198 | Zbl

[41] Nekhoroshev N. N., “Strong stability of an approximate fundamental mode of a nonlinear string equation”, Trans. Moscow Math. Soc., 2002, 151–217 | MR | Zbl

[42] Nekhoroshev N. N., “The Poincaré – Lyapunov – Liouville – Arnol'd theorem”, Funct. Anal. Appl., 28:2 (1994), 128–129 | DOI | MR | Zbl

[43] Nekhoroshev N. N., “Types of integrability on a submanifold and generalizations of Gordon's theorem”, Trans. Moscow Math. Soc., 66 (2005), 169–241 | DOI | MR | Zbl

[44] Novikov S. P., Taimanov I. A., Modern geometric structures and fields, Grad. Stud. Math., 71, AMS, Providence, R.I., 2006, xx+633 pp. | DOI | MR | Zbl

[45] Fomenko A. T., Symplectic geometry, Adv. Stud. Contemp. Math., 5, 2nd ed., CRC Press, Boca Raton, Fla., 1995 | MR