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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 |
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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