From compact semi-toric systems to Hamiltonian S^1-spaces

Previous Article
Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior
DCDS Home
This Issue
Next Article
Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation

January 2015, 35(1): 247-281. doi: 10.3934/dcds.2015.35.247

From compact semi-toric systems to Hamiltonian $S^1$-spaces

Sonja Hohloch ^1, , Silvia Sabatini ^2, and Daniele Sepe ^3,

1.	Section de Mathématiques, EPFL, SB MATHGEOM CAG, Station 8, 1015 Lausanne, Switzerland
2.	CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal
3.	Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga, 24020-240 Niteroi, RJ, Brazil

Received December 2013 Revised March 2014 Published August 2014

Abstract
Related Papers

We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian $S^1$-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian $S^1$-action can be extended to an effective Hamiltonian $\mathbb{T}^2$-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.

Keywords: symplectic invariants., symplectic toric manifolds, integrable Hamiltonian systems, Hamiltonian torus actions, semi-toric systems.

Mathematics Subject Classification: Primary: 37J05, 37J35, 53D2.

Citation: Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

References:

[1]	M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[2]	O. Babelon and B. Doucot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: symplectic invariants and monodromy,, preprint, (). doi: 10.1016/j.geomphys.2014.07.011.
[3]	A. V. Bolsinov and A. A. Oshemkov, Singularities of integrable Hamiltonian systems, in Topological Methods in the Theory of Integrable Systems, Camb. Sci. Publ., Cambridge, 2006, 1-67.
[4]	M. Chaperon, Quelques outils de la théorie des actions différentiables, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, 107-108, Soc. Math. France, Paris, 1983, 259-275.
[5]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, (French) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339.
[6]	J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.
[7]	J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69 (1982), 259-269. doi: 10.1007/BF01399506.
[8]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D thesis, University of Stockholm, 1984.
[9]	L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590.
[10]	A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Translated from the Russian by M. V. Tsaplina., Mathematics and its Applications (Soviet Series) 31, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-3069-8.
[11]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933.
[12]	Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc., 672 (1999). doi: 10.1090/memo/0672.
[13]	Y. Karshon, L. Kessler and M. Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom., 5 (2007), 139-166. doi: 10.4310/JSG.2007.v5.n2.a1.
[14]	Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., 353 (2001), 4831-4861 (electronic). doi: 10.1090/S0002-9947-01-02799-4.
[15]	Y. Karshon and S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom., 2 (2003), 25-82. doi: 10.4310/JSG.2004.v2.n1.a2.
[16]	Y. Karshon and S. Tolman, Classification of Hamiltonian torus actions with two dimensional quotients,, , (). doi: 10.2140/gt.2014.18.669.
[17]	F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, 1984. doi: 10.1007/BF01145470.
[18]	N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187. doi: 10.4310/JSG.2010.v8.n2.a2.
[19]	E. Miranda and N. T. Zung, Equivariant normal form for non-degenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Éc. Norm. Sup., 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.
[20]	Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi: 10.1007/s00222-009-0190-x.
[21]	Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi: 10.1007/s11511-011-0060-4.
[22]	Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455. doi: 10.1090/S0273-0979-2011-01338-6.
[23]	D. A. Sadovskií and B. I. Zĥilinskií, Monodromy, diabolic points and angular momentum coupling, Phys. Lett. A., 256 (1999), 235-244. doi: 10.1016/S0375-9601(99)00229-7.
[24]	M. Symington, Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA), Proc. Sympos. Pure Math., 71, Amer. Math. Soc., 2003, 153-208. doi: 10.1090/pspum/071/2024634.
[25]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.
[26]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004.
[27]	J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.
[28]	N. T. Zung, Symplectic topology of integrable Hamiltonian systems I: Arnol'd-Liouville with singularities, Compositio Math., 101 (1996), 179-215.
[29]	N. T. Zung, A note on focus-focus singularities, Diff. Geom. Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3.
[30]	N. T. Zung, Another note on focus-focus singularities, Lett. Math. Phys., 60 (2002), 87-99. doi: 10.1023/A:1015761729603.

show all references

References:

[1]	M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[2]	O. Babelon and B. Doucot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: symplectic invariants and monodromy,, preprint, (). doi: 10.1016/j.geomphys.2014.07.011.
[3]	A. V. Bolsinov and A. A. Oshemkov, Singularities of integrable Hamiltonian systems, in Topological Methods in the Theory of Integrable Systems, Camb. Sci. Publ., Cambridge, 2006, 1-67.
[4]	M. Chaperon, Quelques outils de la théorie des actions différentiables, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, 107-108, Soc. Math. France, Paris, 1983, 259-275.
[5]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, (French) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339.
[6]	J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.
[7]	J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69 (1982), 259-269. doi: 10.1007/BF01399506.
[8]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D thesis, University of Stockholm, 1984.
[9]	L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590.
[10]	A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Translated from the Russian by M. V. Tsaplina., Mathematics and its Applications (Soviet Series) 31, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-3069-8.
[11]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933.
[12]	Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc., 672 (1999). doi: 10.1090/memo/0672.
[13]	Y. Karshon, L. Kessler and M. Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom., 5 (2007), 139-166. doi: 10.4310/JSG.2007.v5.n2.a1.
[14]	Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., 353 (2001), 4831-4861 (electronic). doi: 10.1090/S0002-9947-01-02799-4.
[15]	Y. Karshon and S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom., 2 (2003), 25-82. doi: 10.4310/JSG.2004.v2.n1.a2.
[16]	Y. Karshon and S. Tolman, Classification of Hamiltonian torus actions with two dimensional quotients,, , (). doi: 10.2140/gt.2014.18.669.
[17]	F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, 1984. doi: 10.1007/BF01145470.
[18]	N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187. doi: 10.4310/JSG.2010.v8.n2.a2.
[19]	E. Miranda and N. T. Zung, Equivariant normal form for non-degenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Éc. Norm. Sup., 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.
[20]	Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi: 10.1007/s00222-009-0190-x.
[21]	Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi: 10.1007/s11511-011-0060-4.
[22]	Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455. doi: 10.1090/S0273-0979-2011-01338-6.
[23]	D. A. Sadovskií and B. I. Zĥilinskií, Monodromy, diabolic points and angular momentum coupling, Phys. Lett. A., 256 (1999), 235-244. doi: 10.1016/S0375-9601(99)00229-7.
[24]	M. Symington, Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA), Proc. Sympos. Pure Math., 71, Amer. Math. Soc., 2003, 153-208. doi: 10.1090/pspum/071/2024634.
[25]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.
[26]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004.
[27]	J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.
[28]	N. T. Zung, Symplectic topology of integrable Hamiltonian systems I: Arnol'd-Liouville with singularities, Compositio Math., 101 (1996), 179-215.
[29]	N. T. Zung, A note on focus-focus singularities, Diff. Geom. Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3.
[30]	N. T. Zung, Another note on focus-focus singularities, Lett. Math. Phys., 60 (2002), 87-99. doi: 10.1023/A:1015761729603.

[1]	Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[2]	Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
[3]	P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[4]	George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215
[5]	Sonja Hohloch. Characterization of toric systems via transport costs. Journal of Geometric Mechanics, 2020, 12 (3) : 447-454. doi: 10.3934/jgm.2020027
[6]	Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[7]	Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[8]	Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[9]	Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[10]	Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587
[11]	Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[12]	Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233
[13]	Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[14]	Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
[15]	Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[16]	Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
[17]	C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7
[18]	Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
[19]	Andrew James Bruce, Janusz Grabowski. Symplectic $ {\mathbb Z}_2^n $-manifolds. Journal of Geometric Mechanics, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020
[20]	Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Sonja Hohloch Silvia Sabatini Daniele Sepe
on Google Scholar
Sonja Hohloch Silvia Sabatini Daniele Sepe

[Back to Top]