Kirchhoff's equations of motion via a constrained Zakharov system

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December 2016, 8(4): 461-485. doi: 10.3934/jgm.2016016

Kirchhoff's equations of motion via a constrained Zakharov system

Banavara N. Shashikanth ^1,

1.	Mechanical and Aerospace Engineering Department, MSC 3450, PO Box 30001, New Mexico State University, Las Cruces, NM 88003, United States

Received June 2015 Revised June 2016 Published November 2016

Abstract
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The Kirchhoff problem for a neutrally buoyant rigid body dynamically interacting with an ideal fluid is considered. Instead of the standard Kirchhoff equations, equations of motion in which the pressure terms appear explicitly are considered. These equations are shown to satisfy a Hamiltonian constraint formalism, with the pressure playing the role of the Lagrange multiplier. The constraint is imposed on the shape of a compact fluid surface whose dynamics is governed by the canonical variables introduced by Zakharov for a free-surface. It is also shown that the assumption of neutral buoyancy can be relaxed.

Keywords: Kirchhoff's equations, Zakharov's equations, pressure., Hamiltonian formalism, constraints.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Banavara N. Shashikanth. Kirchhoff's equations of motion via a constrained Zakharov system. Journal of Geometric Mechanics, 2016, 8 (4) : 461-485. doi: 10.3934/jgm.2016016

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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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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