Dual pairs in resonances

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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

Abstract
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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.
[2]	A. Elipe, Complete reduction of oscillators in resonance $p:q$, Phys. Rev. E, 61 (2000), 6477-6484.
[3]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121.
[4]	F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global. Anal. Geom., to appear.
[5]	D. D. Holm, "Geometric Mechanics Part 1: Dynamics and Symmetry," World Scientific, London, 2008.
[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.
[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.
[8]	M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies, Commun. Math. Phys., 48 (1976), 53-79.
[9]	M. Kummer, On resonant classical Hamiltonians with two equal frequencies, Commun. Math. Phys., 58 (1978), 85-112. doi: 10.1007/BF01624789.
[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.
[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.
[12]	J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D, 24 (1987), 391-405.
[13]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323.
[14]	A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.
[15]	J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics (Boston, Mass.), 222 Boston, Birkhäuser, 2004.
[16]	A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.

show all references

References:

[1]	R. Cushman and D. L. Rod, Reduction of the semisimple $1:1$ resonance, Physica D, 6 (1982), 105-112.
[2]	A. Elipe, Complete reduction of oscillators in resonance $p:q$, Phys. Rev. E, 61 (2000), 6477-6484.
[3]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121.
[4]	F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global. Anal. Geom., to appear.
[5]	D. D. Holm, "Geometric Mechanics Part 1: Dynamics and Symmetry," World Scientific, London, 2008.
[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.
[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.
[8]	M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies, Commun. Math. Phys., 48 (1976), 53-79.
[9]	M. Kummer, On resonant classical Hamiltonians with two equal frequencies, Commun. Math. Phys., 58 (1978), 85-112. doi: 10.1007/BF01624789.
[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.
[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.
[12]	J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D, 24 (1987), 391-405.
[13]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323.
[14]	A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.
[15]	J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics (Boston, Mass.), 222 Boston, Birkhäuser, 2004.
[16]	A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.

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