# American Institute of Mathematical Sciences

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

## Dual pairs in resonances

 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.
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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