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.   Google Scholar [2] A. Elipe, Complete reduction of oscillators in resonance $p:q$,, Phys. Rev. E, 61 (2000), 6477.   Google Scholar [3] F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.   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, (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, 232 (2004), 203.   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.   Google Scholar [8] M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies,, Commun. Math. Phys., 48 (1976), 53.   Google Scholar [9] M. Kummer, On resonant classical Hamiltonians with two equal frequencies,, Commun. Math. Phys., 58 (1978), 85.  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.  doi: 10.1512/iumj.1981.30.30022.  Google Scholar [11] M. Kummer, On resonant Hamiltonian systems with finitely many degrees of freedom,, in, 252 (1986), 19.   Google Scholar [12] J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D,, 24 (1987), 24 (1987), 391.   Google Scholar [13] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Phys. D, 7 (1983), 305.   Google Scholar [14] A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems,, Funct. Anal. Appl., 12 (1978), 113.   Google Scholar [15] J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics (Boston, 222 (2004).   Google Scholar [16] A. Weinstein, The local structure of Poisson manifolds,, J. Diff. Geom., 18 (1983), 523.   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.   Google Scholar [2] A. Elipe, Complete reduction of oscillators in resonance $p:q$,, Phys. Rev. E, 61 (2000), 6477.   Google Scholar [3] F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93.   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, (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, 232 (2004), 203.   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.   Google Scholar [8] M. Kummer, On resonant nonlinearly coupled oscillators with two equal frequencies,, Commun. Math. Phys., 48 (1976), 53.   Google Scholar [9] M. Kummer, On resonant classical Hamiltonians with two equal frequencies,, Commun. Math. Phys., 58 (1978), 85.  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.  doi: 10.1512/iumj.1981.30.30022.  Google Scholar [11] M. Kummer, On resonant Hamiltonian systems with finitely many degrees of freedom,, in, 252 (1986), 19.   Google Scholar [12] J. E. Marsden, Generic Bifurcation of Hamiltonian Systems with Symmetry, appendix to Golubitsky and Stewart, Physica D,, 24 (1987), 24 (1987), 391.   Google Scholar [13] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Phys. D, 7 (1983), 305.   Google Scholar [14] A. S. Mishenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems,, Funct. Anal. Appl., 12 (1978), 113.   Google Scholar [15] J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics (Boston, 222 (2004).   Google Scholar [16] A. Weinstein, The local structure of Poisson manifolds,, J. Diff. Geom., 18 (1983), 523.   Google Scholar
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