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Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $
The method of averaging for Poisson connections on foliations and its applications
| 1. | Departamento de Matemáticas, Universidad de Sonora, Blvd. Luis Encinas y Rosales, s/n, Col. Centro, C.P. 83000, Hermosillo, Son., México |
| 2. | Departamento de Física y Matemáticas, Universidad Autónoma de Ciudad Juárez, Av. del Charro no. 450 nte., Col. Partido Romero, C.P. 32310, Ciudad Juárez, Chihuahua, México |
| 3. | Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Niterói 24210-201, Río de Janeiro, Brasil |
On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.
References:
| [1] |
M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54 (2013), 15pp.
doi: 10.1063/1.4817863. |
| [2] |
M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys., 14 (2017), 15pp.
doi: 10.1142/S0219887817500864. |
| [3] |
O. Brahic and R. L. Fernandes,
Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math., 25 (2014), 901-925.
doi: 10.1016/j.indag.2014.07.007. |
| [4] |
O. Brahic and R. L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41–59.
doi: 10.1090/conm/450/08733. |
| [5] |
T. Courant and A. Weinstein, Beyond Poisson structures, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie, Travaux en Cours, 27, Hermann, Paris, 1988, 39–49. |
| [6] |
T. J. Courant,
Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.1090/S0002-9947-1990-0998124-1. |
| [7] |
J.-P. Dufour and A. Wade,
On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786.
doi: 10.1112/S0010437X07003272. |
| [8] |
V. L. Ginzburg,
Momentum mappings and Poisson cohomology, Internat. J. Math., 7 (1996), 329-358.
doi: 10.1142/S0129167X96000207. |
| [9] |
V. L. Ginzburg,
Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., 10 (1999), 977-1010.
doi: 10.1142/S0129167X99000422. |
| [10] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02950-3. |
| [11] |
J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems, Math. Sci. Res. Inst. Publ., 20, Springer, New York, NY, 1991,209–226.
doi: 10.1007/978-1-4613-9719-9_15. |
| [12] |
J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990).
doi: 10.1090/memo/0436. |
| [13] |
R. Montgomery,
The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120 (1988), 269-294.
doi: 10.1007/BF01217966. |
| [14] |
A. Pedroza, E. Velasco-Barreras and Y. Vorobiev,
Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.
doi: 10.1007/s11005-017-1014-3. |
| [15] |
M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 235, Springer, New York, 2007.
doi: 10.1007/978-0-387-49158-5. |
| [16] |
P. Ševera and A. Weinstein,
Poisson geometry with a 3-form background. Noncommutative geometry and string theory, Progr. Theoret. Phys. Suppl., 144 (2001), 145-154.
doi: 10.1143/PTPS.144.145. |
| [17] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8495-2. |
| [18] |
I. Vaisman,
Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1 (2004), 607-637.
doi: 10.1142/S0219887804000307. |
| [19] |
I. Vaisman,
Foliation-coupling Dirac structures, J. Geom. Phys., 56 (2006), 917-938.
doi: 10.1016/j.geomphys.2005.05.007. |
| [20] |
J. A. Vallejo and Y. Vorobiev, Invariant Poisson realizations and the averaging of Dirac structures, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), 20pp.
doi: 10.3842/SIGMA.2014.096. |
| [21] |
Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001,249–274.
doi: 10.4064/bc54-0-14. |
| [22] |
A. Wade,
Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 3 (2008), 207-217.
doi: 10.1007/s10455-007-9079-3. |
| [23] |
M. Wüstner,
A connected Lie group equals the square of the exponential image, J. Lie Theory, 13 (2003), 307-309.
|
show all references
References:
| [1] |
M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54 (2013), 15pp.
doi: 10.1063/1.4817863. |
| [2] |
M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys., 14 (2017), 15pp.
doi: 10.1142/S0219887817500864. |
| [3] |
O. Brahic and R. L. Fernandes,
Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math., 25 (2014), 901-925.
doi: 10.1016/j.indag.2014.07.007. |
| [4] |
O. Brahic and R. L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41–59.
doi: 10.1090/conm/450/08733. |
| [5] |
T. Courant and A. Weinstein, Beyond Poisson structures, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie, Travaux en Cours, 27, Hermann, Paris, 1988, 39–49. |
| [6] |
T. J. Courant,
Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.1090/S0002-9947-1990-0998124-1. |
| [7] |
J.-P. Dufour and A. Wade,
On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786.
doi: 10.1112/S0010437X07003272. |
| [8] |
V. L. Ginzburg,
Momentum mappings and Poisson cohomology, Internat. J. Math., 7 (1996), 329-358.
doi: 10.1142/S0129167X96000207. |
| [9] |
V. L. Ginzburg,
Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., 10 (1999), 977-1010.
doi: 10.1142/S0129167X99000422. |
| [10] |
I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02950-3. |
| [11] |
J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems, Math. Sci. Res. Inst. Publ., 20, Springer, New York, NY, 1991,209–226.
doi: 10.1007/978-1-4613-9719-9_15. |
| [12] |
J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990).
doi: 10.1090/memo/0436. |
| [13] |
R. Montgomery,
The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120 (1988), 269-294.
doi: 10.1007/BF01217966. |
| [14] |
A. Pedroza, E. Velasco-Barreras and Y. Vorobiev,
Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.
doi: 10.1007/s11005-017-1014-3. |
| [15] |
M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 235, Springer, New York, 2007.
doi: 10.1007/978-0-387-49158-5. |
| [16] |
P. Ševera and A. Weinstein,
Poisson geometry with a 3-form background. Noncommutative geometry and string theory, Progr. Theoret. Phys. Suppl., 144 (2001), 145-154.
doi: 10.1143/PTPS.144.145. |
| [17] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8495-2. |
| [18] |
I. Vaisman,
Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1 (2004), 607-637.
doi: 10.1142/S0219887804000307. |
| [19] |
I. Vaisman,
Foliation-coupling Dirac structures, J. Geom. Phys., 56 (2006), 917-938.
doi: 10.1016/j.geomphys.2005.05.007. |
| [20] |
J. A. Vallejo and Y. Vorobiev, Invariant Poisson realizations and the averaging of Dirac structures, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), 20pp.
doi: 10.3842/SIGMA.2014.096. |
| [21] |
Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001,249–274.
doi: 10.4064/bc54-0-14. |
| [22] |
A. Wade,
Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 3 (2008), 207-217.
doi: 10.1007/s10455-007-9079-3. |
| [23] |
M. Wüstner,
A connected Lie group equals the square of the exponential image, J. Lie Theory, 13 (2003), 307-309.
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