A family of compact semitoric systems with two focus-focus singularities

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September 2018, 10(3): 331-357. doi: 10.3934/jgm.2018012

A family of compact semitoric systems with two focus-focus singularities

Sonja Hohloch ^1,, and Joseph Palmer ^2,

1.	University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, B-2020 Antwerpen, Belgium
2.	Rutgers University, Department of Mathematics, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

* Corresponding author: Sonja Hohloch

Received October 2017 Revised June 2018 Published August 2018

Fund Project: The first author was partially supported by the Research Fund of the University of Antwerp and the second author is partially supported by an AMS-Simons travel grant

About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngọc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a $6$-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.

Keywords: Symplectic geometry, integrable system, semitoric integrable system, Hamiltonian system, momentum map, focus-focus singularity.

Mathematics Subject Classification: Primary: 37J35; Secondary: 53D20.

Citation: Sonja Hohloch, Joseph Palmer. A family of compact semitoric systems with two focus-focus singularities. Journal of Geometric Mechanics, 2018, 10 (3) : 331-357. doi: 10.3934/jgm.2018012

References:

[1]	J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics).
[2]	J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849.
[3]	M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[4]	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. doi: 10.1201/9780203643426.
[5]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi. doi: 10.1017/CBO9780511526367.007.
[6]	R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser/Springer, Basel, 2015, URL https://doi.org/10.1007/978-3-0348-0918-4. doi: 10.1007/978-3-0348-0918-4.
[7]	R. Cushman and V. N. San, Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincaré, 3 (2002), 883-894. doi: 10.1007/s00023-002-8640-7.
[8]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339. doi: 10.24033/bsmf.2100.
[9]	H. Dullin and Á. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811. doi: 10.1007/s00332-016-9290-0.
[10]	K. Efstathiou and N. Martynchuk, Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115. doi: 10.1016/j.geomphys.2016.05.014.
[11]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.
[12]	M. Gaudin, Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098. doi: 10.1051/jphys:0197600370100108700.
[13]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933.
[14]	S. Hohloch, S. Sabatini and D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.
[15]	S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages.
[16]	D. M. Kane, J. Palmer and Á. Pelayo, Classifying toric and semitoric fans by lifting equations from $\rm{SL}_2(\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp. doi: 10.3842/SIGMA.2018.016.
[17]	D. M. Kane, J. Palmer and Á. Pelayo, Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74. doi: 10.1016/j.geomphys.2017.12.005.
[18]	Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), ⅷ+71pp. doi: 10.1090/memo/0672.
[19]	Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419.
[20]	Y. Le Floch, Á. Pelayo and S. Vũ Ngọc, Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413. doi: 10.1007/s00208-015-1259-z.
[21]	J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https://doi.org/10.1007/978-0-387-21792-5, A basic exposition of classical mechanical systems. doi: 10.1007/978-0-387-21792-5.
[22]	E. Miranda and N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.
[23]	J. Palmer, Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217. doi: 10.1016/j.geomphys.2017.02.008.
[24]	Á. Pelayo, Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436. doi: 10.1090/bull/1572.
[25]	Á. Pelayo, T. Ratiu and S. Vũ Ngọc, The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028. doi: 10.1088/1361-6544/aa8aec.
[26]	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.
[27]	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.
[28]	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.
[29]	Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154. doi: 10.1007/s00220-011-1360-4.
[30]	M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universitá degli Studi di Roma Tre, 2007.
[31]	T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093.
[32]	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.
[33]	M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208. doi: 10.1090/pspum/071/2024634.
[34]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.
[35]	S. Vũ Ngọc, Systémes Intégrables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synthéses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2006.
[36]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004.
[37]	C. Wacheux, Systémes Intégrables Semi-toriques et Polytopes Moment, PhD thesis, Université de Rennes 1, 2013.
[38]	J. Williamson, On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.

show all references

References:

[1]	J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics).
[2]	J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849.
[3]	M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[4]	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. doi: 10.1201/9780203643426.
[5]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi. doi: 10.1017/CBO9780511526367.007.
[6]	R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser/Springer, Basel, 2015, URL https://doi.org/10.1007/978-3-0348-0918-4. doi: 10.1007/978-3-0348-0918-4.
[7]	R. Cushman and V. N. San, Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincaré, 3 (2002), 883-894. doi: 10.1007/s00023-002-8640-7.
[8]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339. doi: 10.24033/bsmf.2100.
[9]	H. Dullin and Á. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811. doi: 10.1007/s00332-016-9290-0.
[10]	K. Efstathiou and N. Martynchuk, Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115. doi: 10.1016/j.geomphys.2016.05.014.
[11]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.
[12]	M. Gaudin, Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098. doi: 10.1051/jphys:0197600370100108700.
[13]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933.
[14]	S. Hohloch, S. Sabatini and D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.
[15]	S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages.
[16]	D. M. Kane, J. Palmer and Á. Pelayo, Classifying toric and semitoric fans by lifting equations from $\rm{SL}_2(\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp. doi: 10.3842/SIGMA.2018.016.
[17]	D. M. Kane, J. Palmer and Á. Pelayo, Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74. doi: 10.1016/j.geomphys.2017.12.005.
[18]	Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), ⅷ+71pp. doi: 10.1090/memo/0672.
[19]	Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419.
[20]	Y. Le Floch, Á. Pelayo and S. Vũ Ngọc, Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413. doi: 10.1007/s00208-015-1259-z.
[21]	J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https://doi.org/10.1007/978-0-387-21792-5, A basic exposition of classical mechanical systems. doi: 10.1007/978-0-387-21792-5.
[22]	E. Miranda and N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.
[23]	J. Palmer, Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217. doi: 10.1016/j.geomphys.2017.02.008.
[24]	Á. Pelayo, Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436. doi: 10.1090/bull/1572.
[25]	Á. Pelayo, T. Ratiu and S. Vũ Ngọc, The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028. doi: 10.1088/1361-6544/aa8aec.
[26]	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.
[27]	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.
[28]	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.
[29]	Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154. doi: 10.1007/s00220-011-1360-4.
[30]	M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universitá degli Studi di Roma Tre, 2007.
[31]	T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093.
[32]	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.
[33]	M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208. doi: 10.1090/pspum/071/2024634.
[34]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.
[35]	S. Vũ Ngọc, Systémes Intégrables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synthéses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2006.
[36]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004.
[37]	C. Wacheux, Systémes Intégrables Semi-toriques et Polytopes Moment, PhD thesis, Université de Rennes 1, 2013.
[38]	J. Williamson, On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.

Figure 1. An image of the momentum map $(J_{(1, 2)}, H_{(s_1, s_2)})$ with the rank 0 points marked for varying values of $s_1, s_2\in [0, 1]$. Notice that the coupled angular momenta system shown in Figure 3 is the bottom row of the system shown in this figure since the coupled angular momenta is the special case for which $s_2 = 0$

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Figure 2. Left: a plot of the set $\gamma$, which is the union of $\gamma_{(S, N)}$ (blue) and $\gamma_{(N, S)}$ (orange), see Equation (23). Right: Values of $(s_1, s_2)$ for which the system $(J_{(1, 2)}, H_{(s_1, s_2)})$ has focus-focus values at: only the point $(S, N)$ (blue), only the point $(N, S)$ (orange), or at both points (green). The system is degenerate on the black curves. Compare with Figure 1

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Figure 4. Four semitoric polygons associated to the system (1). The slanted edges all have slope $\pm 1$. For each polygon the $x$-coordinates of the vertices, from left to right, are $-R_1-R_2$, $R_1-R_2$, $-R_1+R_2$, and $R_1+R_2$ (since we assume $R_1<R_2$)

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Figure 3. The momentum map image for the coupled angular momenta system with the rank zero points marked. As the coupling parameter $t$ changes one of the rank zero points transitions from being elliptic-elliptic to being focus-focus and then back to elliptic-elliptic

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Figure 5. This figure analyses the right hand side of Equation (22): The plot on the left shows the graph of the right hand side of Equation (22) which is always below $-1.06066$. The contour plot on the right displays the associated level sets

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Figure 6. Case $(e_1, e_2) = (1, 1)$: on the left, the graph of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, 1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, 1))}$ are shown

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Figure 7. Case $(e_1, e_2) = (1, -1)$: on the left, the graph of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, -1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, -1))}$ are shown

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