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September  2012, 4(3): 297-311. doi: 10.3934/jgm.2012.4.297

Dual pairs in resonances

Darryl D. Holm 1, and  Cornelia Vizman 2,

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
2. 

Department of Mathematics, West University of Timişoara, 300223 Timişoara, Romania

Received  November 2010 Revised  May 2011 Published  October 2012

A family of dual pairs of Poisson maps associated to $n:m$ and $n:-m$ resonances are investigated using Nambu-type Poisson structures.
Keywords: Poisson brackets., Dual pairs.
Mathematics Subject Classification: Primary: 53D17, 53D20; Secondary: 70H0.
Citation: Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297
References:
[1]

R. Cushman and D. L. Rod, Reduction of the semisimple $1:1$ resonance, Physica D, 6 (1982), 105-112.  Google Scholar

[2]

A. Elipe, Complete reduction of oscillators in resonance $p:q$, Phys. Rev. E, 61 (2000), 6477-6484. Google Scholar

[3]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121.  Google Scholar

[4]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Global. Anal. Geom., ().   Google Scholar

[5]

D. D. Holm, "Geometric Mechanics Part 1: Dynamics and Symmetry," World Scientific, London, 2008.  Google Scholar

[6]

D. D. Holm and J. E. Marsden [2004], Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, 203-235, Progr. Math., 232, J. E. Marsden and T. S. Ratiu, Editors, Birkhäuser Boston, Boston, MA, 2004. Google Scholar

[7]

T. Iwai, On reduction of two degrees of freedom Hamiltonian systems by an $S^1$ action and $SO_0(1,2)$ as a dynamical group, J. Math. Phys., 26 (1985), 885-893.  Google Scholar

[8]

M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies, Commun. Math. Phys., 48 (1976), 53-79.  Google Scholar

[9]

M. Kummer, On resonant classical Hamiltonians with two equal frequencies, Commun. Math. Phys., 58 (1978), 85-112. doi: 10.1007/BF01624789.  Google Scholar

[10]

M. Kummer, On the construction of the reduced phase space of a Hamiltonian system with symmetry, Indiana Univ. Math. J., 30 (1981), 281-291. doi: 10.1512/iumj.1981.30.30022.  Google Scholar

[11]

M. Kummer, On resonant Hamiltonian systems with finitely many degrees of freedom, in "Local and Global Methods in Nonlinear Dynamics" (edited by A. V. Sáenz), Lecture Notes in Physics, Springer-Verlag, New York, 252 (1986), 19-31.  Google Scholar

[12]

J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D, 24 (1987), 391-405.  Google Scholar

[13]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. Google Scholar

[14]

A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. Google Scholar

[15]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics (Boston, Mass.), 222 Boston, Birkhäuser, 2004.  Google Scholar

[16]

A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.  Google Scholar

show all references

References:
[1]

R. Cushman and D. L. Rod, Reduction of the semisimple $1:1$ resonance, Physica D, 6 (1982), 105-112.  Google Scholar

[2]

A. Elipe, Complete reduction of oscillators in resonance $p:q$, Phys. Rev. E, 61 (2000), 6477-6484. Google Scholar

[3]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121.  Google Scholar

[4]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Global. Anal. Geom., ().   Google Scholar

[5]

D. D. Holm, "Geometric Mechanics Part 1: Dynamics and Symmetry," World Scientific, London, 2008.  Google Scholar

[6]

D. D. Holm and J. E. Marsden [2004], Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, 203-235, Progr. Math., 232, J. E. Marsden and T. S. Ratiu, Editors, Birkhäuser Boston, Boston, MA, 2004. Google Scholar

[7]

T. Iwai, On reduction of two degrees of freedom Hamiltonian systems by an $S^1$ action and $SO_0(1,2)$ as a dynamical group, J. Math. Phys., 26 (1985), 885-893.  Google Scholar

[8]

M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies, Commun. Math. Phys., 48 (1976), 53-79.  Google Scholar

[9]

M. Kummer, On resonant classical Hamiltonians with two equal frequencies, Commun. Math. Phys., 58 (1978), 85-112. doi: 10.1007/BF01624789.  Google Scholar

[10]

M. Kummer, On the construction of the reduced phase space of a Hamiltonian system with symmetry, Indiana Univ. Math. J., 30 (1981), 281-291. doi: 10.1512/iumj.1981.30.30022.  Google Scholar

[11]

M. Kummer, On resonant Hamiltonian systems with finitely many degrees of freedom, in "Local and Global Methods in Nonlinear Dynamics" (edited by A. V. Sáenz), Lecture Notes in Physics, Springer-Verlag, New York, 252 (1986), 19-31.  Google Scholar

[12]

J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D, 24 (1987), 391-405.  Google Scholar

[13]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. Google Scholar

[14]

A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. Google Scholar

[15]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics (Boston, Mass.), 222 Boston, Birkhäuser, 2004.  Google Scholar

[16]

A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.  Google Scholar

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