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Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems. (English) Zbl 1379.53097

The authors of this short note study adiabatic perturbations of slow-fast Hamitonian systems and of some of their generalizations. At the core of the study is the fact that an adiabatic Hamiltonian system can be seen as a perturbed system on a fibre bundle with a “slow” base and a deformation of a “fast” fibrewise Poisson bracket. For the reader’s sake the authors recall the relevent definitions and facts. As they study objects on fibre bundles, Ehresmann connections play an important role in the study. Another ingredient of the study are actions of compact groups on Poisson fibre bundles and their influence on 2-cocycles related to Poisson structures. The authors demonstrate that every horizontal 2-cocycle is cohomologous to its average (cf. Corollary 10). The final section of the paper is dedicated to G-invariant normalizations. They show, among other things, that a generalized slow-fast Hamiltonian system can be transformed into a particular normal form (cf. Proposition 13). General considerations are supplemented by examples.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
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[1] Arnold, V. I., Mathematical Methods of Classical Mechanics, (1978), Springer, New York · Zbl 0386.70001
[2] Arnold, V. I.; Kozlov, V. V.; Neishtad, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3, (1988), Springer-Verlag, Berlin, New York
[3] Neishtadt, A. I.; Criag, W., Hamiltonian Dynamical Systems and Applications, Averaging method and adiabatic invariants, 53-66, (2008), Springer Science+Business Media B.V.
[4] Vorobiev, Yu., The averaging in Hamiltonian systems on slow-fast phase spaces with \(\mathbb{S}^1\)-symmetry, Phys. Atomic Nucl., 74, 1170-1774, (2011)
[5] E. A. Kudryavtseva, Periodic solutions of planetary systems with satellites and the averaging method in systems with slow and fast variables, preprint (2012), arXiv:1201.6356v6.
[6] Avendaño-Camacho, M.; Vorobiev, Yu., On the global structure of normal forms for slow-fast Hamiltonian systems, Russ. J. Math. Phys., 20, 2, 138-148, (2013) · Zbl 1276.70014
[7] Avendaño-Camacho, M.; Vallejo, J. A.; Vorobiev, Yu., Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54, 1-15, (2013) · Zbl 1302.37032
[8] Karasev, M. V., Adiabatic approximation via hodograph translation and zero-curvature equations, Russ. J. Math. Phys., 21, 2, 197-218, (2014) · Zbl 1311.81157
[9] Karasev, M. V., Adiabatics using the phase space translations and small parameter “dynamics”, Russ. J. Math. Phys., 22, 1, 20-25, (2015) · Zbl 1327.81217
[10] L. M. Lerman and E. I. Yakovlev, Geometry of slow-fast Hamiltonian systems and Painlevé equations, preprint (2015), arXiv:1511.08454.
[11] Vorobiev, Yu.; Avendaño-Camacho, M., The averaging method on slow-fast phase spaces with symmetry, J. Phys.: Conf. Ser., 343, 1-11, (2012)
[12] Marsden, J. E.; Montgomery, R.; Ratiu, T., Reduction, Symmetry and Phases in Mechanics, 88, 1-110, (2012), American Mathematical Society Providence, RI · Zbl 0713.58052
[13] Vallejo, J. A.; Vorobiev, Yu., Invariant Poisson realizations and the averaging of Dirac structures, SIGMA, 10, 096, (2014) · Zbl 1301.53089
[14] Montgomery, R., 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, 269-294, (1988) · Zbl 0689.58043
[15] Golin, S.; Knauf, A.; Marmi, S., The hannay angles: geometry, adiabaticity, and an example, Commun. Math. Phys., 123, 95-122, (1989) · Zbl 0825.58012
[16] Kolář, I.; Michor, P. W.; Slovák, J., Natural Operations in Differential Geometry, (1993), Springer, Berlin · Zbl 0782.53013
[17] Vaisman, I., Lectures on the Geometry of Poisson Manifolds, 118, (1994), Birkhauser, Verlag, Basel · Zbl 0810.53019
[18] Vaisman, I., Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1, 5, 607-637, (2004) · Zbl 1079.53130
[19] Vorob’ev, Yu. M.; Karasev, M. V., Poisson manifolds and their Schouten bracket, Funct. Anal. Appl., 22, 1-9, (1988) · Zbl 0667.58018
[20] Dufour, J. P.; Zung, N. T., Poisson Structures and their Normal Forms, (2005), Springer, Basel
[21] Guillemin, V.; Lerman, E.; Sternberg, S., Symplectic Fibrations and Multiplicity Diagrams, (1996), Cambridge University Press, Cambridge · Zbl 0870.58023
[22] Avendaño-Camacho, M.; Vorobiev, Yu., Homological equations for tensor fields and periodic averaging, Russ. J. Math. Phys., 18, 3, 243-257, (2011) · Zbl 1262.37027
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