American Institute of Mathematical Sciences

Advanced
June  2013, 5(2): 233-256. doi: 10.3934/jgm.2013.5.233

Twisted isotropic realisations of twisted Poisson structures

Nicola Sansonetto 1, and  Daniele Sepe 2,

1. 

Dipartimento di Informatica, Università degli Studi di Verona, Ca' Vignal 2, Strada Le Grazie 15, 37134 Verona,, Italy
2. 

CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

Received  July 2012 Revised  March 2013 Published  July 2013

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in [3], this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in [17], which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in [8]. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising some of the main results of [13].
Keywords: almost symplectic manifolds., Twisted Poisson manifolds, isotropic realisations, Hamiltonian integrable systems.
Mathematics Subject Classification: 37J35, 53D15, 53D1.
Citation: Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233
References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 47 (2004).   Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595.  doi: 10.1088/0951-7715/10/3/002.  Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?,, in, 44 (1999), 29.  doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[5]

G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems,, Rep. Math. Phys., 53 (2004), 211.  doi: 10.1016/S0034-4877(04)90013-4.  Google Scholar

[6]

A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem,, J. Geom. Phys., 34 (2000), 121.  doi: 10.1016/S0393-0440(99)00058-3.  Google Scholar

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.  doi: 10.1007/s002200050412.  Google Scholar

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[9]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math. (2), 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.   Google Scholar

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42.  doi: 10.1006/jdeq.2000.3852.  Google Scholar

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.   Google Scholar

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy,, Ann. Physics, 324 (2009), 1953.  doi: 10.1016/j.aop.2009.03.008.  Google Scholar

[15]

J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.   Google Scholar

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.  doi: 10.1007/s10440-005-1139-8.  Google Scholar

[19]

, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().   Google Scholar

[20]

F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system,, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007).  doi: 10.3842/SIGMA.2007.051.  Google Scholar

[21]

M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[22]

W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations,, Proc. Amer. Math. Soc., 86 (1982), 511.  doi: 10.1090/S0002-9939-1982-0671227-8.  Google Scholar

[23]

J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341.  doi: 10.1016/S0393-0440(02)00027-X.  Google Scholar

[25]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363.  doi: 10.1007/0-8176-4419-9_12.  Google Scholar

[26]

C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds,, Int. Math. Res. Not., 8 (2011), 1839.  doi: 10.1093/imrn/rnq130.  Google Scholar

[27]

K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).   Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', Texts in Applied Mathematics, 17 (1994).  doi: 10.1007/978-1-4612-2682-6.  Google Scholar

[29]

A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.   Google Scholar

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003).  doi: 10.1017/CBO9780511615450.  Google Scholar

[31]

N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.   Google Scholar

[32]

J.-S. Park, Topological open p-branes,, in, (2001), 311.  doi: 10.1142/9789812799821_0010.  Google Scholar

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().   Google Scholar

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().   Google Scholar

[35]

, N. Sansonetto and D. Sepe,, in preparation., ().   Google Scholar

[36]

D. Sepe, Almost Lagrangian obstruction,, Diff. Geom. Appl., 29 (2011), 787.  doi: 10.1016/j.difgeo.2011.08.007.  Google Scholar

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101.  doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[40]

D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503.  doi: 10.1007/BF01209025.  Google Scholar

show all references

References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 47 (2004).   Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595.  doi: 10.1088/0951-7715/10/3/002.  Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?,, in, 44 (1999), 29.  doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[5]

G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems,, Rep. Math. Phys., 53 (2004), 211.  doi: 10.1016/S0034-4877(04)90013-4.  Google Scholar

[6]

A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem,, J. Geom. Phys., 34 (2000), 121.  doi: 10.1016/S0393-0440(99)00058-3.  Google Scholar

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.  doi: 10.1007/s002200050412.  Google Scholar

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[9]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math. (2), 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.   Google Scholar

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42.  doi: 10.1006/jdeq.2000.3852.  Google Scholar

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.   Google Scholar

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy,, Ann. Physics, 324 (2009), 1953.  doi: 10.1016/j.aop.2009.03.008.  Google Scholar

[15]

J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.   Google Scholar

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.  doi: 10.1007/s10440-005-1139-8.  Google Scholar

[19]

, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().   Google Scholar

[20]

F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system,, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007).  doi: 10.3842/SIGMA.2007.051.  Google Scholar

[21]

M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[22]

W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations,, Proc. Amer. Math. Soc., 86 (1982), 511.  doi: 10.1090/S0002-9939-1982-0671227-8.  Google Scholar

[23]

J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341.  doi: 10.1016/S0393-0440(02)00027-X.  Google Scholar

[25]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363.  doi: 10.1007/0-8176-4419-9_12.  Google Scholar

[26]

C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds,, Int. Math. Res. Not., 8 (2011), 1839.  doi: 10.1093/imrn/rnq130.  Google Scholar

[27]

K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).   Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', Texts in Applied Mathematics, 17 (1994).  doi: 10.1007/978-1-4612-2682-6.  Google Scholar

[29]

A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.   Google Scholar

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003).  doi: 10.1017/CBO9780511615450.  Google Scholar

[31]

N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.   Google Scholar

[32]

J.-S. Park, Topological open p-branes,, in, (2001), 311.  doi: 10.1142/9789812799821_0010.  Google Scholar

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().   Google Scholar

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().   Google Scholar

[35]

, N. Sansonetto and D. Sepe,, in preparation., ().   Google Scholar

[36]

D. Sepe, Almost Lagrangian obstruction,, Diff. Geom. Appl., 29 (2011), 787.  doi: 10.1016/j.difgeo.2011.08.007.  Google Scholar

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101.  doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[40]

D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503.  doi: 10.1007/BF01209025.  Google Scholar

[1]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003
[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]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[4]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[5]

Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[6]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[7]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[8]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[9]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[10]

Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002
[11]

Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002
[12]

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
[13]

Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587
[14]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[15]

Enoch H. Apaza, Regis Soares. Axiom a systems without sinks and sources on $n$-manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 393-401. doi: 10.3934/dcds.2008.21.393
[16]

Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101
[17]

Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801
[18]

Simon Hochgerner, Luis García-Naranjo. $G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball. Journal of Geometric Mechanics, 2009, 1 (1) : 35-53. doi: 10.3934/jgm.2009.1.35
[19]

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
[20]

Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences