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September  2020, 12(3): 447-454. doi: 10.3934/jgm.2020027

Characterization of toric systems via transport costs

Department of Mathematics, University of Antwerp, Middelheimlaan 1, B-2020, Antwerp, Belgium

Received  September 2019 Revised  June 2020 Published  September 2020 Early access  September 2020

Fund Project: The author was partially supported by the FWO-EoS project G0H4518N and the UA-BOF project with Antigoon-ID 31722

We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$ T $ map where $ T\in \mathbb{R}^n $ is the period of the acting $ n $-torus.

Citation: Sonja Hohloch. Characterization of toric systems via transport costs. Journal of Geometric Mechanics, 2020, 12 (3) : 447-454. doi: 10.3934/jgm.2020027
References:
[1]

J. AlonsoH. R. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, J. Geom. Phys., 140 (2019), 131-151.  doi: 10.1016/j.geomphys.2018.09.022.

[2]

J. AlonsoH. R. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, Nonlinearity, 33 (2020), 417-468.  doi: 10.1088/1361-6544/ab4e05.

[3]

J. Alonso and S. Hohloch, Survey on recent developments in semitoric systems, Conference Proceedings of RIMS Kokyuroku 2019, (Research Institute for Mathematical Sciences, Kyoto University, Japan, journal identifier ISSN 1880-2818), no. 2137, 15p., see also arXiv: 1901.10433.

[4]

L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 131–140, Higher Ed. Press, Beijing, 2002.

[5]

L. Ambrosio, Lecture notes on optimal transport problems, In Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics, Lecture Notes in Math., Springer Verlag, 1812 2003, 1–52. doi: 10.1007/978-3-540-39189-0_1.

[6]

L. Ambrosio and N. Gigli, A user's guide to optimal transport. Modelling and optimisation of flows on networks, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2062 2013, 1–155. doi: 10.1007/978-3-642-32160-3_1.

[7]

M. Audin, A. Cannas da Silva and E. Lerman, Symplectic geometry of integrable Hamiltonian systems, Lectures delivered at the Euro Summer School held in Barcelona, July 10-15, 2001., Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2003. x+225 pp. doi: 10.1007/978-3-0348-8071-8.

[8]

O. Babelon and B. Douçot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: Symplectic invariants and monodromy, J. Geom. Phys., 87 (2015), 3-29.  doi: 10.1016/j.geomphys.2014.07.011.

[9]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Translated from the 1999 Russian original. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+730 pp. doi: 10.1201/9780203643426.

[10]

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001. xii+217 pp. doi: 10.1007/978-3-540-45330-7.

[11]

A. De Meulenaere and S. Hohloch, A family of semitoric systems with four focus-focus singularities and two double pinched tori, Preprint arXiv: 1911.11883.

[12]

T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment. (French. English summary) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339.  doi: 10.24033/bsmf.2100.

[13]

H. R. 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.

[14]

H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.

[15]

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals–-elliptic case, Comment. Math. Helv., 65 (1990), 4-35.  doi: 10.1007/BF02566590.

[16]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161.  doi: 10.1007/BF02392620.

[17]

S. Hohloch and J. Palmer, A family of compact semitoric systems with two focus-focus singularities, J. Geom. Mech., 10 (2018), 331-357.  doi: 10.3934/jgm.2018012.

[18]

S. HohlochS. Sabatini and D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.  doi: 10.3934/dcds.2015.35.247.

[19]

L. Kantorovich, On the translocation of masses, C.R. (Doklady) Acad. Sci. URSS (N.S.), 37 (1942), 199-201. 

[20]

L. Kantorovič, On a problem of Monge (in Russian), Uspekhi Mat. Nauk., 3, (1948), 225–226.

[21]

Y. Karshon, Periodic Hamiltonian Flows on Four-Dimensional Manifolds, Mem. Amer. Math. Soc., 141 1999, no. 672, viii+71 pp. doi: 10.1090/memo/0672.

[22]

Y. Le Floch and J. Palmer, Semitoric families, Preprint arXiv: 1810.06915.

[23]

Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, J. Nonlinear Sci., 29 (2019), 655-708.  doi: 10.1007/s00332-018-9501-y.

[24]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Third edition. Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017. xi+623 pp. doi: 10.1093/oso/9780198794899.001.0001.

[25]

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.

[26]

G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année (1781), 666–704.

[27]

J. Palmer, Á. Pelayo and X. Tang, Semitoric systems of non-simple type, Preprint arXiv: 1909.03501.

[28]

A. 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.

[29]

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.

[30]

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.

[31]

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.

[32]

S. Rachev and L. Rüschendorf, Mass Transportation Problems, Probability and its Applications, Springer I + II.

[33]

F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. xxvii+353 pp. doi: 10.1007/978-3-319-20828-2.

[34]

V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math., 141 (1979) 1–178.

[35]

M. Thorpe, Introduction to Optimal Transport, cf., http://www.math.cmu.edu/ mthorpe/OTNotes

[36]

C. Villani, Optimal Transport: Old and New, Springer 2009. doi: 10.1007/978-3-540-71050-9.

[37]

S. Vũ Ngọc and C. Wacheux, Smooth normal forms for integrable Hamiltonian systems near a focus-focus singularity, Acta Math. Vietnam., 38 (2013), 107-122.  doi: 10.1007/s40306-013-0012-5.

show all references

References:
[1]

J. AlonsoH. R. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, J. Geom. Phys., 140 (2019), 131-151.  doi: 10.1016/j.geomphys.2018.09.022.

[2]

J. AlonsoH. R. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, Nonlinearity, 33 (2020), 417-468.  doi: 10.1088/1361-6544/ab4e05.

[3]

J. Alonso and S. Hohloch, Survey on recent developments in semitoric systems, Conference Proceedings of RIMS Kokyuroku 2019, (Research Institute for Mathematical Sciences, Kyoto University, Japan, journal identifier ISSN 1880-2818), no. 2137, 15p., see also arXiv: 1901.10433.

[4]

L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 131–140, Higher Ed. Press, Beijing, 2002.

[5]

L. Ambrosio, Lecture notes on optimal transport problems, In Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics, Lecture Notes in Math., Springer Verlag, 1812 2003, 1–52. doi: 10.1007/978-3-540-39189-0_1.

[6]

L. Ambrosio and N. Gigli, A user's guide to optimal transport. Modelling and optimisation of flows on networks, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2062 2013, 1–155. doi: 10.1007/978-3-642-32160-3_1.

[7]

M. Audin, A. Cannas da Silva and E. Lerman, Symplectic geometry of integrable Hamiltonian systems, Lectures delivered at the Euro Summer School held in Barcelona, July 10-15, 2001., Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2003. x+225 pp. doi: 10.1007/978-3-0348-8071-8.

[8]

O. Babelon and B. Douçot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: Symplectic invariants and monodromy, J. Geom. Phys., 87 (2015), 3-29.  doi: 10.1016/j.geomphys.2014.07.011.

[9]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Translated from the 1999 Russian original. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+730 pp. doi: 10.1201/9780203643426.

[10]

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001. xii+217 pp. doi: 10.1007/978-3-540-45330-7.

[11]

A. De Meulenaere and S. Hohloch, A family of semitoric systems with four focus-focus singularities and two double pinched tori, Preprint arXiv: 1911.11883.

[12]

T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment. (French. English summary) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339.  doi: 10.24033/bsmf.2100.

[13]

H. R. 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.

[14]

H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.

[15]

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals–-elliptic case, Comment. Math. Helv., 65 (1990), 4-35.  doi: 10.1007/BF02566590.

[16]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161.  doi: 10.1007/BF02392620.

[17]

S. Hohloch and J. Palmer, A family of compact semitoric systems with two focus-focus singularities, J. Geom. Mech., 10 (2018), 331-357.  doi: 10.3934/jgm.2018012.

[18]

S. HohlochS. Sabatini and D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.  doi: 10.3934/dcds.2015.35.247.

[19]

L. Kantorovich, On the translocation of masses, C.R. (Doklady) Acad. Sci. URSS (N.S.), 37 (1942), 199-201. 

[20]

L. Kantorovič, On a problem of Monge (in Russian), Uspekhi Mat. Nauk., 3, (1948), 225–226.

[21]

Y. Karshon, Periodic Hamiltonian Flows on Four-Dimensional Manifolds, Mem. Amer. Math. Soc., 141 1999, no. 672, viii+71 pp. doi: 10.1090/memo/0672.

[22]

Y. Le Floch and J. Palmer, Semitoric families, Preprint arXiv: 1810.06915.

[23]

Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, J. Nonlinear Sci., 29 (2019), 655-708.  doi: 10.1007/s00332-018-9501-y.

[24]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Third edition. Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017. xi+623 pp. doi: 10.1093/oso/9780198794899.001.0001.

[25]

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.

[26]

G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année (1781), 666–704.

[27]

J. Palmer, Á. Pelayo and X. Tang, Semitoric systems of non-simple type, Preprint arXiv: 1909.03501.

[28]

A. 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.

[29]

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.

[30]

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.

[31]

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.

[32]

S. Rachev and L. Rüschendorf, Mass Transportation Problems, Probability and its Applications, Springer I + II.

[33]

F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. xxvii+353 pp. doi: 10.1007/978-3-319-20828-2.

[34]

V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math., 141 (1979) 1–178.

[35]

M. Thorpe, Introduction to Optimal Transport, cf., http://www.math.cmu.edu/ mthorpe/OTNotes

[36]

C. Villani, Optimal Transport: Old and New, Springer 2009. doi: 10.1007/978-3-540-71050-9.

[37]

S. Vũ Ngọc and C. Wacheux, Smooth normal forms for integrable Hamiltonian systems near a focus-focus singularity, Acta Math. Vietnam., 38 (2013), 107-122.  doi: 10.1007/s40306-013-0012-5.

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