Hamiltonian structures for projectable dynamics on symplectic fiber bundles

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

Hamiltonian structures for projectable dynamics on symplectic fiber bundles

Guillermo Dávila-Rascón ^1, and Yuri Vorobiev ^1,

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

Received April 2011 Revised October 2011 Published October 2012

Abstract
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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.

Keywords: Ehresmann connection, Poisson tensor., invariant connection, symplectic bundle, coupling, symplectic connection, Hamiltonization problem, projectable dynamics.

Mathematics Subject Classification: Primary: 53D05, 53D17; Secondary: 70H06, 70H0.

Citation: Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete and 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.
[2]	V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Second Edition, Springer-Verlag, 1989.
[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.
[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.
[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.
[6]	G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting, Aportaciones Matemáticas, Memorias Sociedad Matemática Mexicana, 38 (2008), 11-21.
[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.
[8]	G. Dávila-Rascón and Y. Vorobiev, A Hamiltonian approach for skew-product dynamical systems, Russian J. of Math. Phys., 15 (2008), 35-44.
[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.
[10]	F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems, Bol. Soc. Mat. Mexicana, 6 (2000), 213-234.
[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.
[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.
[13]	W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'' Vol. II, Academic Press, New York-London, 1973.
[14]	V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'' Cambridge, Univ. Press, 1984.
[15]	V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'' Cambridge Univ. Press., Cambridge, 1996. doi: 10.1017/CBO9780511574788.
[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.
[17]	V. V. Kozlov, "Symmetries, Topology, and Resonances in Hamiltonian Mechanics,'' Springer-Verlag, 1996. doi: 10.1007/978-3-642-78393-7.
[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.
[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.
[20]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Spinger-Verlag, N. Y., 1994.
[21]	D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'' Oxford Mathematical Monographs, Clarendon Press, Oxford, Second Edition, 1998.
[22]	P. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, Providence, R. I., 2008.
[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.
[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.
[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.
[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.
[27]	Y. M. Vorobiev, Hamiltonian structures of the first variation equations, Sbornik: Mathematics, 191 (2000), 447-502.
[28]	Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie Algebroids, Banach Center Publ., Warzawa, 54 (2001) 249-274.
[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.
[30]	A. Weinstein, "Lectures on Symplectic Manifolds,'' CBMS Lecture Notes 29, Providence, R.I., AMS, 1977.
[31]	N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'' Clarendon Press, Oxford, 1996.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundation of Mechanics,'' Second Edition , Addison Wesley, 1978.
[2]	V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Second Edition, Springer-Verlag, 1989.
[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.
[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.
[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.
[6]	G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting, Aportaciones Matemáticas, Memorias Sociedad Matemática Mexicana, 38 (2008), 11-21.
[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.
[8]	G. Dávila-Rascón and Y. Vorobiev, A Hamiltonian approach for skew-product dynamical systems, Russian J. of Math. Phys., 15 (2008), 35-44.
[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.
[10]	F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems, Bol. Soc. Mat. Mexicana, 6 (2000), 213-234.
[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.
[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.
[13]	W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'' Vol. II, Academic Press, New York-London, 1973.
[14]	V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'' Cambridge, Univ. Press, 1984.
[15]	V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'' Cambridge Univ. Press., Cambridge, 1996. doi: 10.1017/CBO9780511574788.
[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.
[17]	V. V. Kozlov, "Symmetries, Topology, and Resonances in Hamiltonian Mechanics,'' Springer-Verlag, 1996. doi: 10.1007/978-3-642-78393-7.
[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.
[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.
[20]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Spinger-Verlag, N. Y., 1994.
[21]	D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'' Oxford Mathematical Monographs, Clarendon Press, Oxford, Second Edition, 1998.
[22]	P. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, Providence, R. I., 2008.
[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.
[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.
[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.
[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.
[27]	Y. M. Vorobiev, Hamiltonian structures of the first variation equations, Sbornik: Mathematics, 191 (2000), 447-502.
[28]	Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie Algebroids, Banach Center Publ., Warzawa, 54 (2001) 249-274.
[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.
[30]	A. Weinstein, "Lectures on Symplectic Manifolds,'' CBMS Lecture Notes 29, Providence, R.I., AMS, 1977.
[31]	N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'' Clarendon Press, Oxford, 1996.

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