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On monodromy in integrable Hamiltonian systems
Martynchuk, Nikolay
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[1] V. I. Arnold, A theorem of Liouville concerning integrable problems of dy-namics, Sibirsk. Mat. Zh. (1963), no. 2, 471–474.
[2] V. I. Arnol’d, Mathematical methods of classical mechanics, 2 ed., Graduate
Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989, Translated by K. Vogtmann and A. Weinstein.
[3] V.I. Arnol’d and A. Avez, Ergodic problems of classical mechanics, W.A.
Benjamin, Inc., 1968.
[4] M. Audin, Torus actions on symplectic manifolds, Birkh¨auser, 2004.
[5] L. Bates and R. Cushman, Scattering monodromy and the A1 singularity,
Central European Journal of Mathematics 5 (2007), no. 3, 429–451.
[6] L.M. Bates, Monodromy in the champagne bottle, Journal of Applied
Math-ematics and Physics (ZAMP) 42 (1991), no. 6, 837–847.
[7] L.M. Bates and F. Fass`o, No monodromy in the champagne bottle, or
singu-larities of a superintegrable system, J. Geom. Mech. 8 (2016), no. 4, 375–389.
[8] L.M. Bates and M. Zou, Degeneration of hamiltonian monodromy cycles,
Nonlinearity 6 (1993), no. 2, 313–335.
[9] S. Bochner, Compact groups of differentiable transformations, Annals of
Mathematics 46 (1945), no. 3, 372–381.
[10] A.V. Bolsinov, Izosimov A.M., A.Y. Konyaev, and A.A. Oshemkov, Algebra
and topology of integrable systems. Research problems (in Russian), Trudy
Sem. Vektor. Tenzor. Anal. 28 (2012), 119–191.
[11] A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems:
Geom-etry, Topology, Classification, CRC Press, 2004.
[12] A.V. Bolsinov, S.V. Matveev, and Fomenko A.T., Topological classification of integrable Hamiltonian systems with two degrees of freedom. list of systems of small complexity, Russian Mathematical Surveys 45 (1990), no. 2, 59.
114 BIBLIOGRAPHY
[13] M. Born, Zur Quantenmechanik der Stoßvorg¨ange, Zeitschrift f¨ur Physik 37
(1926), 863–867.
[14] H.W. Broer, K. Efstathiou, and O.V. Lukina, A geometric fractional
mon-odromy theorem, Discrete and Continuous Dynamical Systems 3 (2010), no. 4, 517–532.
[15] C.L. Charlier, Die Mechanik des Himmels, Veit and Comp, 1902.
[16] J.M. Cook, Banach algebras and asymptotic mechanics, Application of
Mathematics to Problems in Theoretical Physics: Proceedings, Summer School of Theoretical Physics, vol. 6, 1967, pp. 209–246.
[17] R. Cushman and J.J. Duistermaat, The quantum mechanical spherical
pen-dulum, Bulletin of the American Mathematical Society 19 (1988), no. 2, 475–479.
[18] R. H. Cushman and D. A. Sadovski´ı, Monodromy in the hydrogen atom in
crossed fields, Physica D: Nonlinear Phenomena 142 (2000), no. 1-2, 166– 196.
[19] R. H. Cushman and S. V˜u Ngo.c, Sign of the monodromy for Liouville
inte-grable systems, Annales Henri Poincar´e 3 (2002), no. 5, 883–894.
[20] R.H. Cushman and L.M. Bates, Global aspects of classical integrable systems, 2 ed., Birkh¨auser, 2015.
[21] R.H. Cushman and J.J. Duistermaat, Non-hamiltonian monodromy, Journal of Differential Equations 172 (2001), no. 1, 42–58.
[22] R.H. Cushman and H. Kn¨orrer, The energy momentum mapping of the La-grange top, Differential Geometric Methods in Mathematical Physics, Lec-ture Notes in Mathematics, vol. 1139, Springer, 1985, pp. 12–24.
[23] J.B. Delos, G. Dhont, D.A. Sadovski´ı, and B.I. Zhilinski´ı, Dynamical
mani-festation of hamiltonian monodromy, EPL (Europhysics Letters) 83 (2008),
no. 2, 24003.
[24] T. Delzant, Hamiltoniens p´eriodiques et images convexes de l’application moment, Bulletin de la S. M. F. 116 (1988), no. 3, 315–339.
[25] A. Deprit, Le probleme des deux centres fixes, Bull. Soc. Math. Belg 14 (1962), no. 11, 12–45.
[26] J. Derezinski and C. Gerard, Scattering theory of classical and quantum n-particle systems, Theoretical and Mathematical Physics, Springer Berlin Heidelberg, 2013.
[27] J. J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33 (1980), no. 6, 687–706.
[28] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology
of the symplectic form of the reduced phase space, Inventiones mathematicae
69(1982), no. 2, 259–268.
[29] J.J. Duistermaat, The monodromy in the Hamiltonian Hopf bifurcation,
Zeitschrift f¨ur Angewandte Mathematik und Physik (ZAMP) 49 (1998),
no. 1, 156.
[30] H. Dullin and H. Waalkens, Nonuniqueness of the phase shift in central
scattering due to monodromy, Phys. Rev. Lett. 101 (2008).
[31] H. R. Dullin and R. Montgomery, Syzygies in the two center problem,
Non-linearity 29 (2016), no. 4, 1212.
[32] H. R. Dullin and ´A. Pelayo, Generating hyperbolic singularities in semitoric
systems via Hopf bifurcations, Journal of Nonlinear Science 26 (2016), no. 3, 787–811.
[33] Holger R. Dullin and Holger Waalkens, Defect in the joint spectrum of
hy-drogen due to monodromy, Phys. Rev. Lett. 120 (2018), 020507.
[34] K. Efstathiou, Metamorphoses of Hamiltonian systems with symmetries, Springer, Berlin Heidelberg New York, 2005.
[35] K. Efstathiou and H.W. Broer, Uncovering fractional monodromy, Commu-nications in Mathematical Physics 324 (2013), no. 2, 549–588.
[36] K. Efstathiou, R.H. Cushman, and D.A. Sadovski, Fractional monodromy in the 1:2 resonance, Advances in Mathematics 209 (2007), no. 1, 241 – 273. [37] K. Efstathiou and A. Giacobbe, The topology associated to cusp singular
points, Nonlinearity 25 (2012), no. 12, 3409–3422.
[38] K. Efstathiou, A. Giacobbe, P. Mardeˇsi´c, and D. Sugny, Rotation forms and local hamiltonian monodromy, Journal of Mathematical Physics 58 (2017), no. 2, 022902.
[39] K. Efstathiou, H. Hanßmann, and A. Marchesiello, Bifurcations and
mon-odromy of the axially symmetric1:1:−2 resonance, in preparation.
[40] K. Efstathiou and N. Martynchuk, Monodromy of Hamiltonian systems with complexity-1 torus actions, Geometry and Physics 115 (2017), 104–115. [41] H.A. Erikson and E.L. Hill, A note on the one-electron states of diatomic
116 BIBLIOGRAPHY
[42] L. Euler, Probleme. Un corps ´etant attir´e en raison r´eciproque quarr´ee des
distances vers deux points fixes donn´es, trouver les cas o`u la courbe d´ecrite
par ce corps sera alg´ebrique, Histoire de L’Acad´emie Royale des sciences et
Belles-lettres XVI ((1760), 1767), 228–249.
[43] , De motu corporis ad duo centra virium fixa attracti, Novi
Commen-tarii academiae scientiarum Petropolitanae 10 (1766), 207–242.
[44] , De motu corporis ad duo centra virium fixa attracti, Novi
Commen-tarii academiae scientiarum Petropolitanae 11 (1767), 152–184.
[45] H. Flaschka, A remark on integrable Hamiltonian systems, Physics Letters
A 131 (1988), no. 9, 505 – 508.
[46] A.T. Fomenko, Morse theory of integrable Hamiltonian systems, Dokl. Akad.
Nauk SSSR, vol. 287, 1986, pp. 1071–1075.
[47] , The topology of surfaces of constant energy in integrable
Hamilto-nian systems, and obstructions to integrability, Izvestiya: Mathematics 29 (1987), no. 3, 629–658.
[48] A.T. Fomenko and S.V. Matveev, Algorithmic and computer methods for
three-manifolds, 1st ed., Springer Netherlands, 1997.
[49] A.T. Fomenko and H. Zieschang, Topological invariant and a criterion for equivalence of integrable Hamiltonian systems with two degrees of freedom, Izv. Akad. Nauk SSSR, Ser. Mat. 54 (1990), no. 3, 546–575 (Russian). [50] I.A. Gerasimov, Euler problem of two fixed centers, Friazino, 2007, in
Rus-sian.
[51] A. Giacobbe, Fractional monodromy: Parallel transport of homology cycles, Diff. Geom. and Appl. 26 (2008), 140–150.
[52] A. Hatcher, Notes on basic 3-manifold topology, Available online, 2000. [53] I.W. Herbst, Classical scattering with long range forces, Comm. Math. Phys.
35(1974), no. 3, 193–214.
[54] W. Hunziker, The S-matrix in classical mechanics, Comm. Math. Phys. 8 (1968), no. 4, 282–299.
[55] C. G. J. Jacobi, Vorlesungen ¨˜ uber Dynamik, Chelsea Publ., New York, 1884.
[56] M. Jankins and W. D. Neumann, Lectures on seifert manifolds, vol. 2, Bran-deis University, 1983.
[57] C. Jung, Connection between conserved quantities of the Hamiltonian and of the S-matrix, Journal of Physics A: Mathematical and General 26 (1993), no. 5, 1091.
[58] Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus ac-tions, Transactions of the American Mathematical Society 353 (2001), 4831– 4861.
[59] S. Kim, Homoclinic orbits in the Euler problem of two fixed centers,
https://arxiv.org/abs/1606.05622 (2017).
[60] M. Klein and A. Knauf, Classical planar scattering by coulombic potentials,
Lecture Notes in Physics Monographs, Springer Berlin Heidelberg, 2008.
[61] A. Knauf, Qualitative aspects of classical potential scattering, Regul. Chaotic
Dyn. 4 (1999), no. 1, 3–22.
[62] , Mathematische Physik, Springer-Lehrbuch Masterclass, Springer
Berlin Heidelberg, 2011.
[63] A. Knauf and M. Krapf, The non-trapping degree of scattering, Nonlinearity
21(2008), no. 9, 2023.
[64] A. Knauf and M. Seri, Symbolic dynamics of magnetic bumps, Regular and
Chaotic Dynamics 22 (2017), no. 4, 448–454.
[65] E.A. Kudryavtseva and T.A. Lepskii, The topology of Lagrangian foliations
of integrable systems with hyperelliptic Hamiltonian, Sbornik: Mathematics
202(2011), no. 3, 373.
[66] J.L. Lagrange, Miscellania taurinensia, Recherches sur la mouvement d’un corps qui est attir´e vers deux centres fixes 14 (1766-69).
[67] L.M. Lerman and Ya.L. Umanski˘ı, Classification of four-dimensional
inte-grable Hamiltonian systems and Poisson actions of R2 in extended
neigh-borhoods of simple singular points. i, Russian Academy of Sciences. Sbornik
Mathematics 77 (1994), no. 2, 511–542.
[68] J. Liouville, Note sur l’int´egration des ´equations diff´erentielles de la
Dy-namique, pr´esent´ee au Bureau des Longitudes le 29 juin 1853., Journal de
Math´ematiques pures et appliqu´ees (1855), 137–138.
[69] O. V. Lukina, F. Takens, and H. W. Broer, Global properties of integrable hamiltonian systems, Regular and Chaotic Dynamics 13 (2008), no. 6, 602– 644.
[70] N. Martynchuk, H.R. Dullin, K. Efstathiou, and H. Waalkens, Scattering invariants in euler’s two-center problem, arXiv:1801.09613 (2018).
[71] N. Martynchuk and K. Efstathiou, Parallel transport along Seifert manifolds and fractional monodromy, Communications in Mathematical Physics 356 (2017), no. 2, 427–449.
118 BIBLIOGRAPHY
[72] N. Martynchuk and H. Waalkens, Knauf ’s degree and monodromy in planar
potential scattering, Regular and Chaotic Dynamics 21 (2016), no. 6, 697– 706.
[73] N.N. Martynchuk, Semi-local Liouville equivalence of complex Hamiltonian
systems defined by rational Hamiltonian, Topology and its Applications 191 (2015), no. Supplement C, 119 – 130.
[74] Y Matsumoto, Topology of torus fibrations, Sugaku expositions 2 (1989),
55–73.
[75] V.S. Matveev, Integrable Hamiltonian system with two degrees of freedom.
the topological structure of saturated neighbourhoods of points of focus-focus and saddle-saddle type, Sbornik: Mathematics 187 (1996), no. 4, 495–524.
[76] S. Meesters, Monodromy in the two-center problem, 2017.
[77] J. Milnor, Morse theory, Princeton Univ. Press, Princeton, N. J., 1963.
[78] N.N. Nekhoroshev, Fractional monodromy in the case of arbitrary
reso-nances, Sbornik: Mathematics 198 (2007), no. 3, 383–424.
[79] N.N. Nekhoroshev, D.A. Sadovski´ı, and B.I. Zhilinski´ı, Fractional
Hamilto-nian monodromy, Annales Henri Poincar´e 7 (2006), 1099–1211.
[80] K.F. Niessen, Zur Quantentheorie des Wasserstoffmolek¨ulions, Annalen der
Physik 375 (1923), no. 2, 129–134.
[81] D. ´O’Math´una, Integrable systems in celestial mechanics, Birkh¨auser, Basel,
2008.
[82] W. Pauli, ¨Uber das Modell des Wasserstoffmolek¨ulions, Annalen der Physik
373(1922), no. 11, 177–240.
[83] Alvaro Pelayo and San V˜´ u Ngc, Hamiltonian dynamics and spectral theory
for spin–oscillators, Communications in Mathematical Physics 309 (2012), no. 1, 123–154.
[84] M. M. Postnikov, Differential geometry IV, MIR, 1982.
[85] D.A. Sadovsk´ı, Nekhoroshev’s approach to hamiltonian monodromy, Regular and Chaotic Dynamics 21 (2016), no. 6, 720–758.
[86] S. Schmidt and H.R. Dullin, Dynamics near the p : −q resonance, Physica D: Nonlinear Phenomena 239 (2010), no. 19, 1884–1891.
[87] D. Sepe, S. Sabatini, and S. Hohloch, From Compact Semi-Toric Systems To Hamiltonian S-1-Spaces, 35 (2014), 247–281.
[88] M. Seri, The problem of two fixed centers: bifurcation diagram for positive energies, Journal of Mathematical Physics 56 (2015), no. 1, 012902.
[89] M. Seri, A. Knauf, M. D. Esposti, and T. Jecko, Resonances in the
two-center coulomb systems, Reviews in Mathematical Physics 28 (2016), no. 07, 1650016.
[90] B. Simon, Wave operators for classical particle scattering, Comm. Math.
Phys. 23 (1971), no. 1, 37–48.
[91] D. Sugny, P. Mardeˇsi´c, M. Pelletier, A. Jebrane, and H.R. Jauslin,
Frac-tional Hamiltonian monodromy from a Gauss–Manin monodromy, Journal of Mathematical Physics 49 (2008), no. 4, 042701.
[92] F. Takens, Private communication, (2010).
[93] D.I. Tonkonog, A simple proof of the geometric fractional monodromy
theo-rem, Moscow University Mathematics Bulletin 68 (2013), no. 2, 118–121.
[94] H.K. Urbantke, The Hopf fibration—seven times in physics, Journal of
Ge-ometry and Physics 46 (2003), no. 2, 125–150.
[95] S. V˜u Ngo.c, Quantum monodromy in integrable systems, Communications
in Mathematical Physics 203 (1999), no. 2, 465–479.
[96] , Bohr-Sommerfeld conditions for integrable systems with critical
manifolds of focus-focus type, Communications on Pure and Applied Math-ematics 53 (2000), no. 2, 143–217.
[97] S. V˜u Ngo.c, Moment polytopes for symplectic manifolds with monodromy,
Advances in Mathematics 208 (2007), no. 2, 909 – 934.
[98] T.G. Vosmischera, Integrable systems of celestial mechanics in space of
con-stant curvature, Springer Netherlands, 2003.
[99] H. Waalkens, H.R. Dullin, and P.H. Richter, The problem of two fixed
cen-ters: bifurcations, actions, monodromy, Physica D: Nonlinear Phenomena
196(2004), no. 3-4, 265–310.
[100] , The problem of two fixed centers: bifurcations, actions, monodromy,
Physica D 196 (2004), no. 3-4, 265–310.
[101] Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), no. 2, 127 – 150.
[102] Alan Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics 6 (1971), no. 3, 329 – 346.
120 BIBLIOGRAPHY [103] E.T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies; with an introduction to the problem of three bodies, Cambridge, Uni-versity Press, 1917.
[104] O.A. Zagryadskii, E.A. Kudryavtseva, and D.A. Fedoseev, A generalization of Bertrand’s theorem to surfaces of revolution, Sbornik: Mathematics 203 (2012), no. 8, 1112.
[105] N.T. Zung, A note on focus-focus singularities, Differential Geometry and its Applications 7 (1997), no. 2, 123–130.
[106] , Another note on focus-focus singularities, Letters in Mathematical
