# American Institute of Mathematical Sciences

March  2013, 33(3): 1077-1088. doi: 10.3934/dcds.2013.33.1077

## Hamiltonian structures for projectable dynamics on symplectic fiber bundles

 1 Department of Mathematics, University of Sonora, Hermosillo, C.P. 83000, Mexico, Mexico

Received  April 2011 Revised  October 2011 Published  October 2012

The Hamiltonization problem for projectable vector fields on general symplectic fiber bundles is studied. Necessary and sufficient conditions for the existence of Hamiltonian structures in the class of compatible symplectic structures are derived in terms of invariant symplectic connections. In the case of a flat symplectic bundle, we show that this criterion leads to the study of the solvability of homological type equations.
Citation: Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
##### References:
 [1] R. Abraham and J. E. Marsden, "Foundation of Mechanics,'' Second Edition , Addison Wesley, 1978. Google Scholar [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Second Edition, Springer-Verlag, 1989.  Google Scholar [3] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,'' Encyclopedia of Math. Sci., Vol. 3 (Dynamical Systems III), Springer-Verlag, Berlin-New York, 1988.  Google Scholar [4] O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems I. Incompatible poisson structures, Comm. Math. Phys., 180 (1996), 529-586. doi: 10.1007/BF02099623.  Google Scholar [5] A. D. Bryuno, Normalization of a Hamiltonian system near an invariant cycle or torus, Russian Math. Surveys, 44 (1989), 53-89. doi: 10.1070/RM1989v044n02ABEH002041.  Google Scholar [6] G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting, Aportaciones Matemáticas, Memorias Sociedad Matemática Mexicana, 38 (2008), 11-21.  Google Scholar [7] G. Dávila-Rascón, R. Flores-Espinoza and Y. Vorobiev, Euler equations on $\mathfrak so (4)$ as a nearly integrable Hamiltonian system, Qualitative Theory of Dynamical Systems, 7 (2008), 129-146. doi: 10.1007/s12346-008-0007-0.  Google Scholar [8] G. Dávila-Rascón and Y. Vorobiev, A Hamiltonian approach for skew-product dynamical systems, Russian J. of Math. Phys., 15 (2008), 35-44.  Google Scholar [9] G. Dávila-Rascón and Y. Vorobiev, The first step normalization for Hamiltonian systems with two degrees of freedom over orbit cylinders, Electronic J. of Diff. Equations, 2009 (2009), 1-17.  Google Scholar [10] F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems, Bol. Soc. Mat. Mexicana, 6 (2000), 213-234.  Google Scholar [11] M. Gotay, R. Lashof, J. Sniatycki and A. Weinstein, Closed forms on symplectic fiber bundles, Comment. Math. Helv., 58 (1983), 617-621. doi: 10.1007/BF02564656.  Google Scholar [12] S. Golin, A. Knauf and S. Marmi, The Hannay angles: Geometry, adiabaticity and an example, Commun. Math. Phys., 123 (1989), 95-122. doi: 10.1007/BF01244019.  Google Scholar [13] W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'' Vol. II, Academic Press, New York-London, 1973.  Google Scholar [14] V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'' Cambridge, Univ. Press, 1984.  Google Scholar [15] V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'' Cambridge Univ. Press., Cambridge, 1996. doi: 10.1017/CBO9780511574788.  Google Scholar [16] M. V. Karasev and Yu. M. Vorobjev, Adapted connections, Hamilton dynamics, geometric phases and quantization over isotropic submanifolds, Amer. Math. Soc. Transl. (2), 187 (1998), 203-326.  Google Scholar [17] V. V. Kozlov, "Symmetries, Topology, and Resonances in Hamiltonian Mechanics,'' Springer-Verlag, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar [18] J. E. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics, Memoirs of the AMS, Providence, R. I., 88 (1990), 1-110.  Google Scholar [19] J. R. Marsden, T. S. Ratiu and G. Raugel, Symplectic connections and the linearization of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, Sect. A, 117 (1991), 329-380.  Google Scholar [20] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Spinger-Verlag, N. Y., 1994.  Google Scholar [21] D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'' Oxford Mathematical Monographs, Clarendon Press, Oxford, Second Edition, 1998.  Google Scholar [22] P. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, Providence, R. I., 2008.  Google Scholar [23] R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures, Cont. Math. AMS, (Boulder Proceedings on Fluids and Plasmas), 28 (1984), 101-114.  Google Scholar [24] 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.  Google Scholar [25] A. Neishtadt, Averaging method and adiabatic invariants, in "Hamiltonian Dynamical Systems and Applications'' (ed. W. Craig), Springer Science + Business Media B. V., 2008, 53-66.  Google Scholar [26] S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci., U.S.A., 74 (1977), 5253-5254. doi: 10.1073/pnas.74.12.5253.  Google Scholar [27] Y. M. Vorobiev, Hamiltonian structures of the first variation equations, Sbornik: Mathematics, 191 (2000), 447-502. Google Scholar [28] Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie Algebroids, Banach Center Publ., Warzawa, 54 (2001) 249-274.  Google Scholar [29] Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds, Amer. Math. Soc. Transl., AMS, Providence, R. I., 216 (2005), 137-239.  Google Scholar [30] A. Weinstein, "Lectures on Symplectic Manifolds,'' CBMS Lecture Notes 29, Providence, R.I., AMS, 1977.  Google Scholar [31] N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'' Clarendon Press, Oxford, 1996.  Google Scholar

show all references

##### References:
 [1] R. Abraham and J. E. Marsden, "Foundation of Mechanics,'' Second Edition , Addison Wesley, 1978. Google Scholar [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Second Edition, Springer-Verlag, 1989.  Google Scholar [3] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,'' Encyclopedia of Math. Sci., Vol. 3 (Dynamical Systems III), Springer-Verlag, Berlin-New York, 1988.  Google Scholar [4] O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems I. Incompatible poisson structures, Comm. Math. Phys., 180 (1996), 529-586. doi: 10.1007/BF02099623.  Google Scholar [5] A. D. Bryuno, Normalization of a Hamiltonian system near an invariant cycle or torus, Russian Math. Surveys, 44 (1989), 53-89. doi: 10.1070/RM1989v044n02ABEH002041.  Google Scholar [6] G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting, Aportaciones Matemáticas, Memorias Sociedad Matemática Mexicana, 38 (2008), 11-21.  Google Scholar [7] G. Dávila-Rascón, R. Flores-Espinoza and Y. Vorobiev, Euler equations on $\mathfrak so (4)$ as a nearly integrable Hamiltonian system, Qualitative Theory of Dynamical Systems, 7 (2008), 129-146. doi: 10.1007/s12346-008-0007-0.  Google Scholar [8] G. Dávila-Rascón and Y. Vorobiev, A Hamiltonian approach for skew-product dynamical systems, Russian J. of Math. Phys., 15 (2008), 35-44.  Google Scholar [9] G. Dávila-Rascón and Y. Vorobiev, The first step normalization for Hamiltonian systems with two degrees of freedom over orbit cylinders, Electronic J. of Diff. Equations, 2009 (2009), 1-17.  Google Scholar [10] F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems, Bol. Soc. Mat. Mexicana, 6 (2000), 213-234.  Google Scholar [11] M. Gotay, R. Lashof, J. Sniatycki and A. Weinstein, Closed forms on symplectic fiber bundles, Comment. Math. Helv., 58 (1983), 617-621. doi: 10.1007/BF02564656.  Google Scholar [12] S. Golin, A. Knauf and S. Marmi, The Hannay angles: Geometry, adiabaticity and an example, Commun. Math. Phys., 123 (1989), 95-122. doi: 10.1007/BF01244019.  Google Scholar [13] W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'' Vol. II, Academic Press, New York-London, 1973.  Google Scholar [14] V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'' Cambridge, Univ. Press, 1984.  Google Scholar [15] V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'' Cambridge Univ. Press., Cambridge, 1996. doi: 10.1017/CBO9780511574788.  Google Scholar [16] M. V. Karasev and Yu. M. Vorobjev, Adapted connections, Hamilton dynamics, geometric phases and quantization over isotropic submanifolds, Amer. Math. Soc. Transl. (2), 187 (1998), 203-326.  Google Scholar [17] V. V. Kozlov, "Symmetries, Topology, and Resonances in Hamiltonian Mechanics,'' Springer-Verlag, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar [18] J. E. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics, Memoirs of the AMS, Providence, R. I., 88 (1990), 1-110.  Google Scholar [19] J. R. Marsden, T. S. Ratiu and G. Raugel, Symplectic connections and the linearization of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, Sect. A, 117 (1991), 329-380.  Google Scholar [20] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Spinger-Verlag, N. Y., 1994.  Google Scholar [21] D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'' Oxford Mathematical Monographs, Clarendon Press, Oxford, Second Edition, 1998.  Google Scholar [22] P. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, Providence, R. I., 2008.  Google Scholar [23] R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures, Cont. Math. AMS, (Boulder Proceedings on Fluids and Plasmas), 28 (1984), 101-114.  Google Scholar [24] 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.  Google Scholar [25] A. Neishtadt, Averaging method and adiabatic invariants, in "Hamiltonian Dynamical Systems and Applications'' (ed. W. Craig), Springer Science + Business Media B. V., 2008, 53-66.  Google Scholar [26] S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci., U.S.A., 74 (1977), 5253-5254. doi: 10.1073/pnas.74.12.5253.  Google Scholar [27] Y. M. Vorobiev, Hamiltonian structures of the first variation equations, Sbornik: Mathematics, 191 (2000), 447-502. Google Scholar [28] Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie Algebroids, Banach Center Publ., Warzawa, 54 (2001) 249-274.  Google Scholar [29] Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds, Amer. Math. Soc. Transl., AMS, Providence, R. I., 216 (2005), 137-239.  Google Scholar [30] A. Weinstein, "Lectures on Symplectic Manifolds,'' CBMS Lecture Notes 29, Providence, R.I., AMS, 1977.  Google Scholar [31] N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'' Clarendon Press, Oxford, 1996.  Google Scholar
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