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Hamiltonian structures for projectable dynamics on symplectic fiber bundles
1. | Department of Mathematics, University of Sonora, Hermosillo, C.P. 83000, Mexico, Mexico |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,'', Second Edition, (1978). Google Scholar |
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics,'', Second Edition, (1989).
|
[3] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,'', Encyclopedia of Math. Sci., (1988).
|
[4] |
O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems I. Incompatible poisson structures,, Comm. Math. Phys., 180 (1996), 529.
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.
doi: 10.1070/RM1989v044n02ABEH002041. |
[6] |
G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting,, Aportaciones Matemáticas, 38 (2008), 11.
|
[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.
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.
|
[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.
|
[10] |
F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems,, Bol. Soc. Mat. Mexicana, 6 (2000), 213.
|
[11] |
M. Gotay, R. Lashof, J. Sniatycki and A. Weinstein, Closed forms on symplectic fiber bundles,, Comment. Math. Helv., 58 (1983), 617.
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.
doi: 10.1007/BF01244019. |
[13] |
W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'', Vol. II, (1973).
|
[14] |
V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'', Cambridge, (1984).
|
[15] |
V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'', Cambridge Univ. Press., (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.
|
[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, 88 (1990), 1.
|
[19] |
J. R. Marsden, T. S. Ratiu and G. Raugel, Symplectic connections and the linearization of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 117 (1991), 329.
|
[20] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'', Spinger-Verlag, (1994).
|
[21] |
D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'', Oxford Mathematical Monographs, (1998).
|
[22] |
P. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, (2008).
|
[23] |
R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Cont. Math. AMS, 28 (1984), 101.
|
[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.
doi: 10.1007/BF01217966. |
[25] |
A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.
|
[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., 74 (1977), 5253.
doi: 10.1073/pnas.74.12.5253. |
[27] |
Y. M. Vorobiev, Hamiltonian structures of the first variation equations,, Sbornik: Mathematics, 191 (2000), 447. Google Scholar |
[28] |
Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf,, Lie Algebroids, 54 (2001), 249.
|
[29] |
Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds,, Amer. Math. Soc. Transl., 216 (2005), 137.
|
[30] |
A. Weinstein, "Lectures on Symplectic Manifolds,'', CBMS Lecture Notes 29, (1977).
|
[31] |
N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'', Clarendon Press, (1996).
|
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,'', Second Edition, (1978). Google Scholar |
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics,'', Second Edition, (1989).
|
[3] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,'', Encyclopedia of Math. Sci., (1988).
|
[4] |
O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems I. Incompatible poisson structures,, Comm. Math. Phys., 180 (1996), 529.
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.
doi: 10.1070/RM1989v044n02ABEH002041. |
[6] |
G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting,, Aportaciones Matemáticas, 38 (2008), 11.
|
[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.
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.
|
[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.
|
[10] |
F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems,, Bol. Soc. Mat. Mexicana, 6 (2000), 213.
|
[11] |
M. Gotay, R. Lashof, J. Sniatycki and A. Weinstein, Closed forms on symplectic fiber bundles,, Comment. Math. Helv., 58 (1983), 617.
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.
doi: 10.1007/BF01244019. |
[13] |
W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'', Vol. II, (1973).
|
[14] |
V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'', Cambridge, (1984).
|
[15] |
V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'', Cambridge Univ. Press., (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.
|
[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, 88 (1990), 1.
|
[19] |
J. R. Marsden, T. S. Ratiu and G. Raugel, Symplectic connections and the linearization of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 117 (1991), 329.
|
[20] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'', Spinger-Verlag, (1994).
|
[21] |
D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'', Oxford Mathematical Monographs, (1998).
|
[22] |
P. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, (2008).
|
[23] |
R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Cont. Math. AMS, 28 (1984), 101.
|
[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.
doi: 10.1007/BF01217966. |
[25] |
A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.
|
[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., 74 (1977), 5253.
doi: 10.1073/pnas.74.12.5253. |
[27] |
Y. M. Vorobiev, Hamiltonian structures of the first variation equations,, Sbornik: Mathematics, 191 (2000), 447. Google Scholar |
[28] |
Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf,, Lie Algebroids, 54 (2001), 249.
|
[29] |
Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds,, Amer. Math. Soc. Transl., 216 (2005), 137.
|
[30] |
A. Weinstein, "Lectures on Symplectic Manifolds,'', CBMS Lecture Notes 29, (1977).
|
[31] |
N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'', Clarendon Press, (1996).
|
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