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Semi-global symplectic invariants of the Euler top
Twisted isotropic realisations of twisted Poisson structures
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 |
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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.
doi: 10.1007/s002200050412. |
[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. |
[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. |
[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. |
[11] |
M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.
|
[12] |
R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42.
doi: 10.1006/jdeq.2000.3852. |
[13] |
P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.
|
[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. |
[15] |
J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.
doi: 10.1002/cpa.3160330602. |
[16] |
A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.
|
[17] |
F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).
doi: 10.1063/1.2783937. |
[18] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.
doi: 10.1007/s10440-005-1139-8. |
[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. |
[21] |
M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185.
doi: 10.1090/S0002-9947-1980-0561832-4. |
[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. |
[23] |
J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.
doi: 10.1088/0951-7715/8/4/003. |
[24] |
C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341.
doi: 10.1016/S0393-0440(02)00027-X. |
[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. |
[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. |
[27] |
K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).
|
[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. |
[29] |
A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.
|
[30] |
I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003).
doi: 10.1017/CBO9780511615450. |
[31] |
N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.
|
[32] |
J.-S. Park, Topological open p-branes,, in, (2001), 311.
doi: 10.1142/9789812799821_0010. |
[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. |
[37] |
P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145.
doi: 10.1143/PTPS.144.145. |
[38] |
I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994).
doi: 10.1007/978-3-0348-8495-2. |
[39] |
A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101.
doi: 10.1090/S0273-0979-1987-15473-5. |
[40] |
D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503.
doi: 10.1007/BF01209025. |
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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.
doi: 10.1007/s002200050412. |
[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. |
[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. |
[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. |
[11] |
M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.
|
[12] |
R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42.
doi: 10.1006/jdeq.2000.3852. |
[13] |
P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.
|
[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. |
[15] |
J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.
doi: 10.1002/cpa.3160330602. |
[16] |
A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.
|
[17] |
F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).
doi: 10.1063/1.2783937. |
[18] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.
doi: 10.1007/s10440-005-1139-8. |
[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. |
[21] |
M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185.
doi: 10.1090/S0002-9947-1980-0561832-4. |
[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. |
[23] |
J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.
doi: 10.1088/0951-7715/8/4/003. |
[24] |
C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341.
doi: 10.1016/S0393-0440(02)00027-X. |
[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. |
[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. |
[27] |
K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).
|
[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. |
[29] |
A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.
|
[30] |
I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003).
doi: 10.1017/CBO9780511615450. |
[31] |
N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.
|
[32] |
J.-S. Park, Topological open p-branes,, in, (2001), 311.
doi: 10.1142/9789812799821_0010. |
[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. |
[37] |
P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145.
doi: 10.1143/PTPS.144.145. |
[38] |
I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994).
doi: 10.1007/978-3-0348-8495-2. |
[39] |
A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101.
doi: 10.1090/S0273-0979-1987-15473-5. |
[40] |
D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503.
doi: 10.1007/BF01209025. |
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