Skip to main content
SpringerLink
Log in

Cotangent Models for Integrable Systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on b-Poisson/b-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle theorems in these settings: the theorem of Liouville–Mineur–Arnold for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds (Laurent- Gengoux et al., IntMath Res Notices IMRN 8: 1839–1869, 2011). Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a b-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved by the authors and Scott (Math. Pures Appl. (9) 105(1):66–85, 2016). This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the b-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on b-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I.: Mathematical methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, Moscow, (1974) 2nd edn. XVI (1989)

  2. Besse A.: Manifolds All of Whose Geodesics are Closed, vol. 93. Springer, Berlin (2012)

    Google Scholar 

  3. Bieberbach L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297–336 (1911)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bolsinov A.V., Izosimov A.: Singularities of bi-hamiltonian systems. Commun. Math. Phys. 331(2), 507–543 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras Berkeley Mathematics Lecture Notes Series. American Mathematical Society, Providence (1999)

  6. Duistermaat J.J.: On global action-angle coordinates. Commun. Pure Appl. Math. 23, 687–706 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eliasson L.H.: Normal forms for hamiltonian systems with poisson commuting integrals—elliptic case. Comment. Math. Helv. 65(1), 4–35 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gualtieri, M., Li, S., Pelayo, A., Ratiu, T.: Tropical moment maps for toric log symplectic manifolds. arXiv:1407.3300 (2014)

  9. Guillemin V., Miranda E., Pires A.: Codimension one symplectic foliations and regular poisson structures. Bull. Braz. Math. Soc. (N.S.) 42(4), 607–623 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guillemin V., Miranda E., Pires A.: Symplectic and poisson geometry on b-manifolds. Adv. Math. 264, 864–896 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guillemin V., Miranda E., Pires A.R., Scott G.: Toric actions on b-symplectic manifolds. Int. Math. Res. Notices 2015(14), 5818–5848 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics, 2nd edn. Cambridge University Press, Cambridge, xii+468 pp. ISBN: 0-521-38990-9 (1990)

  13. Guillemin, V., Sternberg, S.: A normal form for the moment map. In: Sternberg, S. (ed.) Differential Geometric Methods in Mathematical Physics, Mathematical Physics Studies, vol. 6. D. Reidel (1984)

  14. Kiesenhofer A., Miranda E., Scott G.: Action-angle variables and a kam theorem for b-poisson manifolds, J. Math. Pures Appl. (9) 105(1), 66–85 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Laurent-Gengoux C., Miranda E., Vanhaecke, P.: Action-angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Notices IMRN 8, 1839–1869 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Laurent-Gengoux C., Miranda E.: Coupling symmetries with Poisson structures. Acta Math. Vietnam. 38(1), 21–32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liouville J.: Note sur l’intégration des équations différentielles de la dynamique, présentée au bureau des longitudes le 29 juin 1853. J. Math. Pures Appl. 20, 137–138 (1855)

    Google Scholar 

  18. Marle C.M.: Modèle d’action hamiltonienne d’un groupe de lie sur une variété symplectique. Rend. Sem. Mat. Univ. Politec. Torino 43(2), 227–251 (1985)

    MathSciNet  MATH  Google Scholar 

  19. Mineur, H.: Réduction des systèmes mécaniques à n degrés de liberté admettant n intégrales premières uniformes en involution aux systèmes à variables séparées. J. Math Pure Appl. IX Ser. 15, 385–389 (1936)

  20. Miranda E.: Integrable systems and group actions. Central Eur. J. Math. 12(2), 240–270 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Miranda, E.: On symplectic linearization of singular Lagrangian foliations, Ph.D. thesis. Universitat de Barcelona (2003) ISBN: 9788469412374

  22. Miranda E., Zung N.T.: Equivariant normal forms for nondegenerate singular orbits of integrable hamiltonian systems. Ann. Sci. Ecole Norm. Sup. 37(6), 819–839 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Nest R., Tsygan B.: Formal deformations of symplectic manifolds with boundary. J. Reine Angew. Math. 481, 27–54 (1996)

    MathSciNet  MATH  Google Scholar 

  24. Swan R.: Vector bundles and projective modules. Trans. Am. Math. Soc. 105(2), 264–277 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  25. Thorpe J.: Parallelizability and flat manifolds. Proc. Am. Math. Soc. 16, 138–142 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  26. Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. (23) 3–4, 379–394 (1997)

  27. Weinstein A.: The local structure of poisson manifolds. J. Differ. Geom. 18(3), 523–557 (1983)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Universitat Politècnica de Catalunya and BGSMath, EPSEB, Avinguda del Doctor Marañón 44-50, Barcelona, Spain

    Anna Kiesenhofer & Eva Miranda

Authors
  1. Anna Kiesenhofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eva Miranda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eva Miranda.

Additional information

Communicated by P. Deift

A. Kiesenhofer and E. Miranda are supported by the Grants Reference Number MTM2015-69135-P (MINECO/FEDER) and Reference Number 2014SGR634 (AGAUR). A. Kiesenhofer is supported by a UPC doctoral grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiesenhofer, A., Miranda, E. Cotangent Models for Integrable Systems. Commun. Math. Phys. 350, 1123–1145 (2017). https://doi.org/10.1007/s00220-016-2720-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2720-x

Search

Navigation