June  2019, 11(2): 225-238. doi: 10.3934/jgm.2019012

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

In honor of Darryl Holm's 70th birthday

Received  February 2018 Revised  October 2018 Published  May 2019

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.

Citation: Tomoki Ohsawa. Dual pairs and regularization of Kummer shapes in resonances. Journal of Geometric Mechanics, 2019, 11 (2) : 225-238. doi: 10.3934/jgm.2019012
References:
[1]

J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, 2014. Google Scholar

[2]

R. C. ChurchillM. 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.  Google Scholar

[3]

H. DullinA. 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.  Google Scholar

[4]

M. Gell-Mann, Symmetries of baryons and mesons, Physical Review, 125 (1962), 1067-1084.  doi: 10.1103/PhysRev.125.1067.  Google Scholar

[5]

M. GolubitskyI. 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.  Google Scholar

[6]

G. Haller, Chaos Near Resonance, Applied Mathematical Sciences. Springer, New York, 1999. doi: 10.1007/978-1-4612-1508-0.  Google Scholar

[7]

D. D. Holm, Geometric Mechanics, Part Ⅰ: Dynamics and Symmetry, Imperial College Press, 2nd edition, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics. American Mathematical Society, 2004. doi: 10.1090/gsm/064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[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.  Google Scholar

[16]

A. Weinstein, The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

show all references

References:
[1]

J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, 2014. Google Scholar

[2]

R. C. ChurchillM. 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.  Google Scholar

[3]

H. DullinA. 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.  Google Scholar

[4]

M. Gell-Mann, Symmetries of baryons and mesons, Physical Review, 125 (1962), 1067-1084.  doi: 10.1103/PhysRev.125.1067.  Google Scholar

[5]

M. GolubitskyI. 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.  Google Scholar

[6]

G. Haller, Chaos Near Resonance, Applied Mathematical Sciences. Springer, New York, 1999. doi: 10.1007/978-1-4612-1508-0.  Google Scholar

[7]

D. D. Holm, Geometric Mechanics, Part Ⅰ: Dynamics and Symmetry, Imperial College Press, 2nd edition, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics. American Mathematical Society, 2004. doi: 10.1090/gsm/064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[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.  Google Scholar

[16]

A. Weinstein, The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

Figure 1.  The Kummer shape is regularized to be the sphere (green), and the reduced dynamics (red) (11) is at the intersection of the sphere and the level set (blue) of the Hamiltonian $h$.
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