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 - A, 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, (1978).   Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,'', Second Edition, (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., (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.  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.  doi: 10.1070/RM1989v044n02ABEH002041.  Google Scholar

[6]

G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting,, Aportaciones Matemáticas, 38 (2008), 11.   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.  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.   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.   Google Scholar

[10]

F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems,, Bol. Soc. Mat. Mexicana, 6 (2000), 213.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'', Vol. II, (1973).   Google Scholar

[14]

V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'', Cambridge, (1984).   Google Scholar

[15]

V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'', Cambridge Univ. Press., (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.   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, 88 (1990), 1.   Google Scholar

[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.   Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'', Spinger-Verlag, (1994).   Google Scholar

[21]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'', Oxford Mathematical Monographs, (1998).   Google Scholar

[22]

P. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, (2008).   Google Scholar

[23]

R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Cont. Math. AMS, 28 (1984), 101.   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.  doi: 10.1007/BF01217966.  Google Scholar

[25]

A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.   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., 74 (1977), 5253.  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.   Google Scholar

[28]

Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf,, Lie Algebroids, 54 (2001), 249.   Google Scholar

[29]

Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds,, Amer. Math. Soc. Transl., 216 (2005), 137.   Google Scholar

[30]

A. Weinstein, "Lectures on Symplectic Manifolds,'', CBMS Lecture Notes 29, (1977).   Google Scholar

[31]

N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'', Clarendon Press, (1996).   Google Scholar

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).   Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,'', Encyclopedia of Math. Sci., (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.  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.  doi: 10.1070/RM1989v044n02ABEH002041.  Google Scholar

[6]

G. Dávila-Rascón, Hamiltonian structures for two frequency systems and the KAM-setting,, Aportaciones Matemáticas, 38 (2008), 11.   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.  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.   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.   Google Scholar

[10]

F. Espinoza and Y. Vorobiev, Hamiltonian formalism for fiberwise linear Hamiltonian dynamical systems,, Bol. Soc. Mat. Mexicana, 6 (2000), 213.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

W. Greub, S. Halperin and R. Vanstone., "Connections, Curvature and Cohomology,'', Vol. II, (1973).   Google Scholar

[14]

V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,'', Cambridge, (1984).   Google Scholar

[15]

V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,'', Cambridge Univ. Press., (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.   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, 88 (1990), 1.   Google Scholar

[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.   Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'', Spinger-Verlag, (1994).   Google Scholar

[21]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,'', Oxford Mathematical Monographs, (1998).   Google Scholar

[22]

P. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, (2008).   Google Scholar

[23]

R. Montgomery, J. E. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Cont. Math. AMS, 28 (1984), 101.   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.  doi: 10.1007/BF01217966.  Google Scholar

[25]

A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.   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., 74 (1977), 5253.  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.   Google Scholar

[28]

Y. M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf,, Lie Algebroids, 54 (2001), 249.   Google Scholar

[29]

Y. M. Vorobjev, Poisson structures and linear Euler systems over symplectic manifolds,, Amer. Math. Soc. Transl., 216 (2005), 137.   Google Scholar

[30]

A. Weinstein, "Lectures on Symplectic Manifolds,'', CBMS Lecture Notes 29, (1977).   Google Scholar

[31]

N. M. J. Woodhouse, "Integrability, Self-Duality and Twistor Theory,'', Clarendon Press, (1996).   Google Scholar

[1]

Roderick S. C. Wong, H. Y. Zhang. On the connection formulas of the third Painlevé transcendent. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 541-560. doi: 10.3934/dcds.2009.23.541

[2]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135

[3]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[4]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[5]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

[6]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001

[7]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[8]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[9]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[10]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409

[11]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[12]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

[13]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003

[14]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[15]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[16]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[17]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021004

[18]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[19]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[20]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]