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Dispersive Lamb systems
Dual pairs for matrix groups
1. | Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom |
2. | Department of Mathematics, West University of Timişoara, RO–300223 Timişoara, Romania |
In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair due to Holm and Marsden, and the ideal fluid dual pair due to Marsden and Weinstein.
References:
[1] |
E. Artin, Geometric Algebra, reprint of the 1957 original, John Wiley & Sons, Inc., New York, 1988.
doi: 10.1002/9781118164518. |
[2] |
C. Balleier and T. Wurzbacher,
On the geometry and quantization of symplectic Howe pairs, Math. Z., 271 (2012), 577-591.
doi: 10.1007/s00209-011-0878-7. |
[3] |
A. Blaom, A Geometric Setting for Hamiltonian Perturbation Theory, Memoirs of the American Mathematical Society, vol. 153, AMS, 2001.
doi: 10.1090/memo/0727. |
[4] |
F. Gay-Balmaz and C. Vizman,
Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.
doi: 10.1007/s10455-011-9267-z. |
[5] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breath of Symplectic and Poisson Geometry (eds. J. E. Marsden and T. S. Ratiu), Progress in Mathematics, vol. 232, Birkhäuser, (2005), 203–235.
doi: 10.1007/0-8176-4419-9_8. |
[6] |
R. Howe,
Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.
doi: 10.1090/S0002-9947-1989-0986027-X. |
[7] |
D. Kazhdan, B. Kostant and S. Sternberg,
Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507.
doi: 10.1002/cpa.3160310405. |
[8] |
S. Kudla, Seesaw dual reductive pairs, in Automorphic Forms of Several Variables (Katata, 1983), (eds. I. Satake and Y. Morita), Progress in Mathematics, 46 (1984), 244–268. |
[9] |
J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 218, Springer, 2013.
doi: 10.1007/978-0-387-21752-9. |
[10] |
E. Lerman, R. Montgomery and R. Sjamaar, Examples of singular reduction, in Symplectic Geometry (Warwick, 1990) (ed. D. Salamon), London Mathematical Society Lecture Note Series, vol. 192, Cambridge University Press, (1993), 127–155.
doi: 10.1017/CBO9780511526343.008. |
[11] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, vol. 35, D. Reidel Publishing Company, 1987.
doi: 10.1007/978-94-009-3807-6. |
[12] |
S. Lie, Theorie der Tranformationsgruppen, (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, 1890. Google Scholar |
[13] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Texts in Applied Mathematics, vol. 17, Springer, 1999.
doi: 10.1007/978-0-387-21792-5. |
[14] |
J. E. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Order in chaos (Los Alamos, N.M., 1982). Phys. D, 7 (1983), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[15] |
J.-P. Ortega,
Singular dual pairs, Differential Geom. Appl., 19 (2003), 61-95.
doi: 10.1016/S0926-2245(03)00015-9. |
[16] |
J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, vol. 222, Birkhäuser, 2004.
doi: 10.1007/978-1-4757-3811-7. |
[17] |
P. Skerritt, The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics, arXiv: 1802.04362. Google Scholar |
[18] |
A. Weinstein,
The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |
[19] |
H. Xu,
An SVD-like matrix decomposition and its applications, Linear Algebra Appl., 368 (2003), 1-24.
doi: 10.1016/S0024-3795(03)00370-7. |
show all references
References:
[1] |
E. Artin, Geometric Algebra, reprint of the 1957 original, John Wiley & Sons, Inc., New York, 1988.
doi: 10.1002/9781118164518. |
[2] |
C. Balleier and T. Wurzbacher,
On the geometry and quantization of symplectic Howe pairs, Math. Z., 271 (2012), 577-591.
doi: 10.1007/s00209-011-0878-7. |
[3] |
A. Blaom, A Geometric Setting for Hamiltonian Perturbation Theory, Memoirs of the American Mathematical Society, vol. 153, AMS, 2001.
doi: 10.1090/memo/0727. |
[4] |
F. Gay-Balmaz and C. Vizman,
Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.
doi: 10.1007/s10455-011-9267-z. |
[5] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breath of Symplectic and Poisson Geometry (eds. J. E. Marsden and T. S. Ratiu), Progress in Mathematics, vol. 232, Birkhäuser, (2005), 203–235.
doi: 10.1007/0-8176-4419-9_8. |
[6] |
R. Howe,
Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.
doi: 10.1090/S0002-9947-1989-0986027-X. |
[7] |
D. Kazhdan, B. Kostant and S. Sternberg,
Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507.
doi: 10.1002/cpa.3160310405. |
[8] |
S. Kudla, Seesaw dual reductive pairs, in Automorphic Forms of Several Variables (Katata, 1983), (eds. I. Satake and Y. Morita), Progress in Mathematics, 46 (1984), 244–268. |
[9] |
J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 218, Springer, 2013.
doi: 10.1007/978-0-387-21752-9. |
[10] |
E. Lerman, R. Montgomery and R. Sjamaar, Examples of singular reduction, in Symplectic Geometry (Warwick, 1990) (ed. D. Salamon), London Mathematical Society Lecture Note Series, vol. 192, Cambridge University Press, (1993), 127–155.
doi: 10.1017/CBO9780511526343.008. |
[11] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, vol. 35, D. Reidel Publishing Company, 1987.
doi: 10.1007/978-94-009-3807-6. |
[12] |
S. Lie, Theorie der Tranformationsgruppen, (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, 1890. Google Scholar |
[13] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Texts in Applied Mathematics, vol. 17, Springer, 1999.
doi: 10.1007/978-0-387-21792-5. |
[14] |
J. E. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Order in chaos (Los Alamos, N.M., 1982). Phys. D, 7 (1983), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[15] |
J.-P. Ortega,
Singular dual pairs, Differential Geom. Appl., 19 (2003), 61-95.
doi: 10.1016/S0926-2245(03)00015-9. |
[16] |
J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, vol. 222, Birkhäuser, 2004.
doi: 10.1007/978-1-4757-3811-7. |
[17] |
P. Skerritt, The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics, arXiv: 1802.04362. Google Scholar |
[18] |
A. Weinstein,
The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |
[19] |
H. Xu,
An SVD-like matrix decomposition and its applications, Linear Algebra Appl., 368 (2003), 1-24.
doi: 10.1016/S0024-3795(03)00370-7. |
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