# American Institute of Mathematical Sciences

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

## 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

Received  June 2015 Revised  June 2016 Published  November 2016

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.
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, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar [2] H. Aref and S. W. Jones, Chaotic motion of a solid through ideal fluid,, Phys. Fluids A, 5 (1993), 3026. doi: 10.1063/1.858712. Google Scholar [3] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics,, volume 125 of series Applied Mathematical Sciences, (1998). Google Scholar [4] T. B. Benjamin, Hamiltonian theory for motions of bubbles in an infinite liquid,, J. Fluid Mech., 181 (1987), 349. doi: 10.1017/S002211208700212X. Google Scholar [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. doi: 10.1070/RD2003v008n04ABEH000257. Google Scholar [6] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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, (1993), 23. Google Scholar [8] P. A. M. Dirac, Lectures on Quantum Mechanics,, Second printing of the 1964 original. Belfer Graduate School of Science Monographs Series, (1964). Google Scholar [9] P. Ehrenfest, Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz,, PhD Thesis, (1904). Google Scholar [10] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics,, Springer, (2007). Google Scholar [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. doi: 10.1098/rspa.1994.0098. Google Scholar [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. doi: 10.1017/S002211209500190X. Google Scholar [13] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar [14] G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit,, Journal für die reine und angewandte Mathematik (Crelle's Journal), 1870 (1870), 237. doi: 10.1515/crll.1870.71.237. Google Scholar [15] J. Koiller, Note on coupled motions of vortices and rigid bodies,, Physics Letters A, 120 (1987), 391. doi: 10.1016/0375-9601(87)90685-2. Google Scholar [16] V. V. Kozlov and D. A. Oniščenko, Nonintegrability of Kirchhoff's equations,, Soviet Math. Dokl., 26 (1982), 495. Google Scholar [17] L. Landweber and C. S. Yih, Forces, moments, and added masses for Rankine bodies,, J. Fluid Mech., 1 (1956), 319. doi: 10.1017/S0022112056000184. Google Scholar [18] N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica, 33 (1997), 331. doi: 10.1016/S0005-1098(96)00176-8. Google Scholar [19] D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems,, Physica D, 18 (1986), 391. doi: 10.1016/0167-2789(86)90207-1. Google Scholar [20] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, volume 17 of series Texts in Applied Mathematics, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [21] L. M. Milne-Thomson, Theoretical Hydrodynamics,, $5^{th}$ edition, (1996). Google Scholar [22] S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, Funktsional Anal. i Prilozhen., 15 (1981), 37. Google Scholar [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. Google Scholar [24] S. M. Ramodanov, Motion of a circular cylinder and $N$ point vortices in a perfect fluid,, Reg. Chaotic Dyn., 7 (2002), 291. doi: 10.1070/RD2002v007n03ABEH000211. Google Scholar [25] P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, (1992). Google Scholar [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. doi: 10.1070/RD2005v010n01ABEH000295. Google Scholar [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. doi: 10.1063/1.1445183. Google Scholar [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. doi: 10.1007/s00162-007-0065-y. Google Scholar [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. doi: 10.1007/BF00913182. Google Scholar

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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, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar [2] H. Aref and S. W. Jones, Chaotic motion of a solid through ideal fluid,, Phys. Fluids A, 5 (1993), 3026. doi: 10.1063/1.858712. Google Scholar [3] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics,, volume 125 of series Applied Mathematical Sciences, (1998). Google Scholar [4] T. B. Benjamin, Hamiltonian theory for motions of bubbles in an infinite liquid,, J. Fluid Mech., 181 (1987), 349. doi: 10.1017/S002211208700212X. Google Scholar [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. doi: 10.1070/RD2003v008n04ABEH000257. Google Scholar [6] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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, (1993), 23. Google Scholar [8] P. A. M. Dirac, Lectures on Quantum Mechanics,, Second printing of the 1964 original. Belfer Graduate School of Science Monographs Series, (1964). Google Scholar [9] P. Ehrenfest, Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz,, PhD Thesis, (1904). Google Scholar [10] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics,, Springer, (2007). Google Scholar [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. doi: 10.1098/rspa.1994.0098. Google Scholar [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. doi: 10.1017/S002211209500190X. Google Scholar [13] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar [14] G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit,, Journal für die reine und angewandte Mathematik (Crelle's Journal), 1870 (1870), 237. doi: 10.1515/crll.1870.71.237. Google Scholar [15] J. Koiller, Note on coupled motions of vortices and rigid bodies,, Physics Letters A, 120 (1987), 391. doi: 10.1016/0375-9601(87)90685-2. Google Scholar [16] V. V. Kozlov and D. A. Oniščenko, Nonintegrability of Kirchhoff's equations,, Soviet Math. Dokl., 26 (1982), 495. Google Scholar [17] L. Landweber and C. S. Yih, Forces, moments, and added masses for Rankine bodies,, J. Fluid Mech., 1 (1956), 319. doi: 10.1017/S0022112056000184. Google Scholar [18] N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica, 33 (1997), 331. doi: 10.1016/S0005-1098(96)00176-8. Google Scholar [19] D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems,, Physica D, 18 (1986), 391. doi: 10.1016/0167-2789(86)90207-1. Google Scholar [20] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, volume 17 of series Texts in Applied Mathematics, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [21] L. M. Milne-Thomson, Theoretical Hydrodynamics,, $5^{th}$ edition, (1996). Google Scholar [22] S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, Funktsional Anal. i Prilozhen., 15 (1981), 37. Google Scholar [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. Google Scholar [24] S. M. Ramodanov, Motion of a circular cylinder and $N$ point vortices in a perfect fluid,, Reg. Chaotic Dyn., 7 (2002), 291. doi: 10.1070/RD2002v007n03ABEH000211. Google Scholar [25] P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, (1992). Google Scholar [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. doi: 10.1070/RD2005v010n01ABEH000295. Google Scholar [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. doi: 10.1063/1.1445183. Google Scholar [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. doi: 10.1007/s00162-007-0065-y. Google Scholar [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. doi: 10.1007/BF00913182. Google Scholar
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