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On strain measures and the geodesic distance to $SO_n$ in the general linear group
Kirchhoff's equations of motion via a constrained Zakharov system
1. | Mechanical and Aerospace Engineering Department, MSC 3450, PO Box 30001, New Mexico State University, Las Cruces, NM 88003, United States |
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, volume 75 in series Applied Mathematical Sciences, $2^{nd}$ edition, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
[2] |
H. Aref and S. W. Jones, Chaotic motion of a solid through ideal fluid, Phys. Fluids A, 5 (1993), 3026-3028.
doi: 10.1063/1.858712. |
[3] |
V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, volume 125 of series Applied Mathematical Sciences, Springer-Verlag, 1998. |
[4] |
T. B. Benjamin, Hamiltonian theory for motions of bubbles in an infinite liquid, J. Fluid Mech., 181 (1987), 349-379.
doi: 10.1017/S002211208700212X. |
[5] |
A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. Chaotic Dyn., 8 (2003), 449-462.
doi: 10.1070/RD2003v008n04ABEH000257. |
[6] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[7] |
E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, in Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, 23-26 May 1993, St John's, Newfoundland, Canada, 159-163. Available online from http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf. |
[8] |
P. A. M. Dirac, Lectures on Quantum Mechanics, Second printing of the 1964 original. Belfer Graduate School of Science Monographs Series, 2. Belfer Graduate School of Science, New York; produced and distributed by Academic Press, Inc., New York, 1967. |
[9] |
P. Ehrenfest, Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz, PhD Thesis, University of Vienna, 1904. |
[10] |
L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007. |
[11] |
A. Galper and T. Miloh, Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field, Proc. Roy. Soc. Lond. A, 446 (1994), 169-193.
doi: 10.1098/rspa.1994.0098. |
[12] |
A. Galper and T. Miloh, Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field, J. Fluid. Mech., 295 (1995), 91-120.
doi: 10.1017/S002211209500190X. |
[13] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511624056. |
[14] |
G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal für die reine und angewandte Mathematik (Crelle's Journal), 1870 (1870), 237-262.
doi: 10.1515/crll.1870.71.237. |
[15] |
J. Koiller, Note on coupled motions of vortices and rigid bodies, Physics Letters A, 120 (1987), 391-395.
doi: 10.1016/0375-9601(87)90685-2. |
[16] |
V. V. Kozlov and D. A. Oniščenko, Nonintegrability of Kirchhoff's equations, Soviet Math. Dokl., 26 (1982), 495-498. |
[17] |
L. Landweber and C. S. Yih, Forces, moments, and added masses for Rankine bodies, J. Fluid Mech., 1 (1956), 319-336.
doi: 10.1017/S0022112056000184. |
[18] |
N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, 33 (1997), 331-346.
doi: 10.1016/S0005-1098(96)00176-8. |
[19] |
D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Physica D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, volume 17 of series Texts in Applied Mathematics, $2^{nd}$ edition, Springer-Verlag, 1999.
doi: 10.1007/978-0-387-21792-5. |
[21] |
L. M. Milne-Thomson, Theoretical Hydrodynamics, $5^{th}$ edition, Dover, New York, 1996. |
[22] |
S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, Funktsional Anal. i Prilozhen., 15 (1981), 37-52, Available online from http://www.mi.ras.ru/ snovikov/70.pdf. |
[23] |
S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a fluid and the extended Lyusternik-Shnirel'man-Morse theory. I, Funktsional Anal. i Prilozhen, 15 (1981), 54-66, Available online from http://www.mi.ras.ru/ snovikov/69.pdf. |
[24] |
S. M. Ramodanov, Motion of a circular cylinder and $N$ point vortices in a perfect fluid, Reg. Chaotic Dyn., 7 (2002), 291-298.
doi: 10.1070/RD2002v007n03ABEH000211. |
[25] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge Univesity Press, 1992. |
[26] |
B. N. Shashikanth, Poisson brackets for the dynamically interacting system of a 2D rigid boundary and $N$ point vortices: The case of arbitrary smooth cylinder shapes, Reg. Chaotic Dyn., 10 (2005), 1-14.
doi: 10.1070/RD2005v010n01ABEH000295. |
[27] |
B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a 2-D rigid cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227.
doi: 10.1063/1.1445183. |
[28] |
B. N. Shashikanth, A. Sheshmani, S. D. Kelly and J. E. Marsden, Hamiltonian structure for a neutrally buoyant rigid body interacting with $N$ vortex rings of arbitrary shape: the case of arbitrary smooth body shape, Theoretical and Computational Fluid Dynamics, 22 (2008), 37-64.
doi: 10.1007/s00162-007-0065-y. |
[29] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. Originally published in Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi, 9 (1968), 86-94.
doi: 10.1007/BF00913182. |
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, volume 75 in series Applied Mathematical Sciences, $2^{nd}$ edition, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
[2] |
H. Aref and S. W. Jones, Chaotic motion of a solid through ideal fluid, Phys. Fluids A, 5 (1993), 3026-3028.
doi: 10.1063/1.858712. |
[3] |
V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, volume 125 of series Applied Mathematical Sciences, Springer-Verlag, 1998. |
[4] |
T. B. Benjamin, Hamiltonian theory for motions of bubbles in an infinite liquid, J. Fluid Mech., 181 (1987), 349-379.
doi: 10.1017/S002211208700212X. |
[5] |
A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. Chaotic Dyn., 8 (2003), 449-462.
doi: 10.1070/RD2003v008n04ABEH000257. |
[6] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[7] |
E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, in Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, 23-26 May 1993, St John's, Newfoundland, Canada, 159-163. Available online from http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf. |
[8] |
P. A. M. Dirac, Lectures on Quantum Mechanics, Second printing of the 1964 original. Belfer Graduate School of Science Monographs Series, 2. Belfer Graduate School of Science, New York; produced and distributed by Academic Press, Inc., New York, 1967. |
[9] |
P. Ehrenfest, Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz, PhD Thesis, University of Vienna, 1904. |
[10] |
L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007. |
[11] |
A. Galper and T. Miloh, Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field, Proc. Roy. Soc. Lond. A, 446 (1994), 169-193.
doi: 10.1098/rspa.1994.0098. |
[12] |
A. Galper and T. Miloh, Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field, J. Fluid. Mech., 295 (1995), 91-120.
doi: 10.1017/S002211209500190X. |
[13] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511624056. |
[14] |
G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal für die reine und angewandte Mathematik (Crelle's Journal), 1870 (1870), 237-262.
doi: 10.1515/crll.1870.71.237. |
[15] |
J. Koiller, Note on coupled motions of vortices and rigid bodies, Physics Letters A, 120 (1987), 391-395.
doi: 10.1016/0375-9601(87)90685-2. |
[16] |
V. V. Kozlov and D. A. Oniščenko, Nonintegrability of Kirchhoff's equations, Soviet Math. Dokl., 26 (1982), 495-498. |
[17] |
L. Landweber and C. S. Yih, Forces, moments, and added masses for Rankine bodies, J. Fluid Mech., 1 (1956), 319-336.
doi: 10.1017/S0022112056000184. |
[18] |
N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, 33 (1997), 331-346.
doi: 10.1016/S0005-1098(96)00176-8. |
[19] |
D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Physica D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, volume 17 of series Texts in Applied Mathematics, $2^{nd}$ edition, Springer-Verlag, 1999.
doi: 10.1007/978-0-387-21792-5. |
[21] |
L. M. Milne-Thomson, Theoretical Hydrodynamics, $5^{th}$ edition, Dover, New York, 1996. |
[22] |
S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, Funktsional Anal. i Prilozhen., 15 (1981), 37-52, Available online from http://www.mi.ras.ru/ snovikov/70.pdf. |
[23] |
S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a fluid and the extended Lyusternik-Shnirel'man-Morse theory. I, Funktsional Anal. i Prilozhen, 15 (1981), 54-66, Available online from http://www.mi.ras.ru/ snovikov/69.pdf. |
[24] |
S. M. Ramodanov, Motion of a circular cylinder and $N$ point vortices in a perfect fluid, Reg. Chaotic Dyn., 7 (2002), 291-298.
doi: 10.1070/RD2002v007n03ABEH000211. |
[25] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge Univesity Press, 1992. |
[26] |
B. N. Shashikanth, Poisson brackets for the dynamically interacting system of a 2D rigid boundary and $N$ point vortices: The case of arbitrary smooth cylinder shapes, Reg. Chaotic Dyn., 10 (2005), 1-14.
doi: 10.1070/RD2005v010n01ABEH000295. |
[27] |
B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a 2-D rigid cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227.
doi: 10.1063/1.1445183. |
[28] |
B. N. Shashikanth, A. Sheshmani, S. D. Kelly and J. E. Marsden, Hamiltonian structure for a neutrally buoyant rigid body interacting with $N$ vortex rings of arbitrary shape: the case of arbitrary smooth body shape, Theoretical and Computational Fluid Dynamics, 22 (2008), 37-64.
doi: 10.1007/s00162-007-0065-y. |
[29] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. Originally published in Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi, 9 (1968), 86-94.
doi: 10.1007/BF00913182. |
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