# American Institute of Mathematical Sciences

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. 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. Google Scholar [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. 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. 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. 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. 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. Google Scholar [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. 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. 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. 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
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 $h$.
 [1] Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55 [2] 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 [3] Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014 [4] François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39 [5] M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151. [6] Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56. [7] Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239 [8] Yvette Kosmann-Schwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165-180. doi: 10.3934/jgm.2012.4.165 [9] Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971 [10] Toshiyuki Ogawa, Takashi Okuda. Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance. Networks & Heterogeneous Media, 2012, 7 (4) : 893-926. doi: 10.3934/nhm.2012.7.893 [11] V. Balaji, P. Barik, D. S. Nagaraj. On degenerations of moduli of Hitchin pairs. Electronic Research Announcements, 2013, 20: 103-108. doi: 10.3934/era.2013.20.105 [12] Kevin Kuo, Phong Luu, Duy Nguyen, Eric Perkerson, Katherine Thompson, Qing Zhang. Pairs trading: An optimal selling rule. Mathematical Control & Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489 [13] R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519 [14] Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10. [15] S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86. [16] Jorge Sotomayor, Michail Zhitomirskii. On pairs of foliations defined by vector fields in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 741-749. doi: 10.3934/dcds.2000.6.741 [17] Robert L. Griess, Jr. and Ching Hung Lam. Rootless pairs of $EE_8$-lattices. Electronic Research Announcements, 2008, 15: 52-61. doi: 10.3934/era.2008.15.52 [18] Adriano Regis Rodrigues, César Castilho, Jair Koiller. Vortex pairs on a triaxial ellipsoid and Kimura's conjecture. Journal of Geometric Mechanics, 2018, 10 (2) : 189-208. doi: 10.3934/jgm.2018007 [19] Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 [20] André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351

2018 Impact Factor: 0.525

## Tools

Article outline

Figures and Tables