• Previous Article
    Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries
  • JGM Home
  • This Issue
  • Next Article
    Some remarks about the centre of mass of two particles in spaces of constant curvature
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

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[1]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031

[2]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001

[3]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281

[4]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160

[5]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[6]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008

[7]

Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021003

[8]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[9]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[10]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134

[11]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[12]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

[13]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[14]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[15]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[16]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[17]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[18]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[19]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[20]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (53)
  • HTML views (117)
  • Cited by (0)

Other articles
by authors

[Back to Top]