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 $h$ .
-
Previous Article
Dispersive Lamb systems
- JGM Home
- This Issue
-
Next Article
Embedding Camassa-Holm equations in incompressible Euler
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 |
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.
Mathematics Subject Classification:
37J15, 53D17, 53D20, 70K30.
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. |
| [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. |
| [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 and 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 and 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 and Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489 |
| [13] |
Menglu Feng, Mei Choi Chiu, Hoi Ying Wong. Pairs trading with illiquidity and position limits. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2991-3009. doi: 10.3934/jimo.2019090 |
| [14] |
R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519 |
| [15] |
Vladimir Lazić, Fanjun Meng. On Nonvanishing for uniruled log canonical pairs. Electronic Research Archive, 2021, 29 (5) : 3297-3308. doi: 10.3934/era.2021039 |
| [16] |
Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10. |
| [17] |
Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139 |
| [18] |
Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056 |
| [19] |
Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037 |
| [20] |
D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure and Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]