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Embedding Camassa-Holm equations in incompressible Euler
Dual pairs and regularization of Kummer shapes in resonances
Department of Mathematical Sciences, The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080-3021, USA |
We present an account of dual pairs and the Kummer shapes for $ n:m $ resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on $ \mathfrak{su}(2)^{*} $ is the standard $ (+) $-Lie-Poisson bracket independent of the values of $ (n,m) $ as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of $ (n,m) $. A similar result holds for $ n:-m $ resonance with a paraboloid and $ \mathfrak{su}(1,1)^{*} $. The result also has a straightforward generalization to multidimensional resonances as well.
References:
[1] |
J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, 2014. |
[2] |
R. C. Churchill, M. Kummer and D. L. Rod,
On averaging, reduction, and symmetry in Hamiltonian systems, Journal of Differential Equations, 49 (1983), 359-414.
doi: 10.1016/0022-0396(83)90003-7. |
[3] |
H. Dullin, A. Giacobbe and R. Cushman,
Monodromy in the resonant swing spring, Physica D: Nonlinear Phenomena, 190 (2004), 15-37.
doi: 10.1016/j.physd.2003.10.004. |
[4] |
M. Gell-Mann,
Symmetries of baryons and mesons, Physical Review, 125 (1962), 1067-1084.
doi: 10.1103/PhysRev.125.1067. |
[5] |
M. Golubitsky, I. Stewart and J. E. Marsden,
Generic bifurcation of Hamiltonian systems with symmetry, Physica D: Nonlinear Phenomena, 24 (1987), 391-405.
doi: 10.1016/0167-2789(87)90087-X. |
[6] |
G. Haller, Chaos Near Resonance, Applied Mathematical Sciences. Springer, New York, 1999.
doi: 10.1007/978-1-4612-1508-0. |
[7] |
D. D. Holm, Geometric Mechanics, Part Ⅰ: Dynamics and Symmetry, Imperial College Press, 2nd edition, 2011. |
[8] |
D. D. Holm and C. Vizman,
Dual pairs in resonances, Journal of Geometric Mechanics, 4 (2012), 297-311.
doi: 10.3934/jgm.2012.4.297. |
[9] |
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, Journal of Mathematical Physics, 26 (1985), 885-893.
doi: 10.1063/1.526544. |
[10] |
A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics. American Mathematical Society, 2004.
doi: 10.1090/gsm/064. |
[11] |
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. |
[12] |
M. Kummer, Lecture 1: On resonant Hamiltonian systems with finitely many degrees of freedom, Local and Global Methods of Nonlinear Dynamics (Silver Spring, Md., 1984), 19-31, Lecture Notes in Phys., 252, Springer, Berlin, 1986.
doi: 10.1007/BFb0018325. |
[13] |
M. Kummer, On resonant Hamiltonians with $n$ frequencies, The Physics of Phase Space (College Park, Md., 1986), 63-65, Lecture Notes in Phys., 278, Springer, Berlin, 1987.
doi: 10.1007/3-540-17894-5_321. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, volume 222 of Progress in Mathematics., Birkhäuser, 2004.
doi: 10.1007/978-1-4757-3811-7. |
[16] |
A. Weinstein,
The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |
show all references
References:
[1] |
J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, 2014. |
[2] |
R. C. Churchill, M. Kummer and D. L. Rod,
On averaging, reduction, and symmetry in Hamiltonian systems, Journal of Differential Equations, 49 (1983), 359-414.
doi: 10.1016/0022-0396(83)90003-7. |
[3] |
H. Dullin, A. Giacobbe and R. Cushman,
Monodromy in the resonant swing spring, Physica D: Nonlinear Phenomena, 190 (2004), 15-37.
doi: 10.1016/j.physd.2003.10.004. |
[4] |
M. Gell-Mann,
Symmetries of baryons and mesons, Physical Review, 125 (1962), 1067-1084.
doi: 10.1103/PhysRev.125.1067. |
[5] |
M. Golubitsky, I. Stewart and J. E. Marsden,
Generic bifurcation of Hamiltonian systems with symmetry, Physica D: Nonlinear Phenomena, 24 (1987), 391-405.
doi: 10.1016/0167-2789(87)90087-X. |
[6] |
G. Haller, Chaos Near Resonance, Applied Mathematical Sciences. Springer, New York, 1999.
doi: 10.1007/978-1-4612-1508-0. |
[7] |
D. D. Holm, Geometric Mechanics, Part Ⅰ: Dynamics and Symmetry, Imperial College Press, 2nd edition, 2011. |
[8] |
D. D. Holm and C. Vizman,
Dual pairs in resonances, Journal of Geometric Mechanics, 4 (2012), 297-311.
doi: 10.3934/jgm.2012.4.297. |
[9] |
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, Journal of Mathematical Physics, 26 (1985), 885-893.
doi: 10.1063/1.526544. |
[10] |
A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics. American Mathematical Society, 2004.
doi: 10.1090/gsm/064. |
[11] |
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. |
[12] |
M. Kummer, Lecture 1: On resonant Hamiltonian systems with finitely many degrees of freedom, Local and Global Methods of Nonlinear Dynamics (Silver Spring, Md., 1984), 19-31, Lecture Notes in Phys., 252, Springer, Berlin, 1986.
doi: 10.1007/BFb0018325. |
[13] |
M. Kummer, On resonant Hamiltonians with $n$ frequencies, The Physics of Phase Space (College Park, Md., 1986), 63-65, Lecture Notes in Phys., 278, Springer, Berlin, 1987.
doi: 10.1007/3-540-17894-5_321. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, volume 222 of Progress in Mathematics., Birkhäuser, 2004.
doi: 10.1007/978-1-4757-3811-7. |
[16] |
A. Weinstein,
The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |

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