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 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.

[1]

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

[2]

Otávio J. N. T. N. dos Santos, Emerson L. Monte Carmelo. A connection between sumsets and covering codes of a module. Advances in Mathematics of Communications, 2018, 12 (3) : 595-605. doi: 10.3934/amc.2018035

[3]

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811

[4]

Shunfu Jin, Wuyi Yue, Zhanqiang Huo. Performance evaluation for connection oriented service in the next generation Internet. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 749-761. doi: 10.3934/naco.2011.1.749

[5]

Pavel I. Etingof. Galois groups and connection matrices of q-difference equations. Electronic Research Announcements, 1995, 1: 1-9.

[6]

Flank D. M. Bezerra, Rodiak N. Figueroa-López, Marcelo J. D. Nascimento. Fractional oscillon equations; solvability and connection with classical oscillon equations. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2257-2277. doi: 10.3934/cpaa.2021067

[7]

Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems and Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012

[8]

Marta Strani. Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1653-1667. doi: 10.3934/cpaa.2014.13.1653

[9]

Roger E. Khayat, Martin Ostoja-Starzewski. On the objective rate of heat and stress fluxes. Connection with micro/nano-scale heat convection. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 991-998. doi: 10.3934/dcdsb.2011.15.991

[10]

Jon Chaika, Donald Robertson. Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang). Journal of Modern Dynamics, 2019, 15: 329-343. doi: 10.3934/jmd.2019023

[11]

Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455

[12]

Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54.

[13]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259

[14]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[15]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583

[16]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321

[17]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

[18]

Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu. Study of adaptive symplectic methods for simulating charged particle dynamics. Journal of Computational Dynamics, 2019, 6 (2) : 429-448. doi: 10.3934/jcd.2019022

[19]

Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37

[20]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (152)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]