Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

Banavara N. Shashikanth

doi:10.3934/jgm.2020003

Previous Article
Rolling and no-slip bouncing in cylinders
JGM Home
This Issue
Next Article
Constraint algorithm for singular field theories in the k-cosymplectic framework

March 2020, 12(1): 25-52. doi: 10.3934/jgm.2020003

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

Banavara N. Shashikanth

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA

Received January 2019 Published January 2020

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.

Keywords: Wave-body coupled dynamics, Momentum laws, Poisson brackets, Lie-Poisson bracket, Zakharov bracket.

Mathematics Subject Classification: Primary: 37K05, 74F10, 76B07; Secondary: 53D17.

Citation: Banavara N. Shashikanth. Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame. Journal of Geometric Mechanics, 2020, 12 (1) : 25-52. doi: 10.3934/jgm.2020003

References:

[1]	G. A. Athanassoulis, Analytical dynamics of wave body interaction, in Mathematical techniques for water waves (ed. B. N. Mandal), Advances in Fluid Mechanics, 8, Computational Mechanics Publications, Southampton, Boston (1997), 79–154. Available from: urlhttps://www.researchgate.net/publication/266301554_Analytical_dynamics_of_wave-body_interaction
[2]	G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, Cambridge, 1999.
[3]	M. Brøns, M. C. Thompson, T. Leweke and K. Hourigan, Vorticity generation and conservation for two-dimensional interfaces and boundaries, J. Fluid Mech., 758 (2014), 63-93.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[5]	A. Compelli, R. Ivanov and M. Todorov, Hamiltonian models for the propogation of irrotational surface gravity waves over a variable bottom, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170091, 15 pp. doi: 10.1098/rsta.2017.0091.
[6]	L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007.
[7]	C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys, Rev. Lett., 19 (1967), 1095-1097.
[8]	F. Gay-Balmaz, J. E. Marsden and T. S. Ratiu, Reduced variational formulations in free boundary continuum mechanics, J. Nonlinear Sci., 22 (2012), 463-497. doi: 10.1007/s00332-012-9143-4.
[9]	R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.
[10]	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.
[11]	F. John, On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57. doi: 10.1002/cpa.3160020102.
[12]	F. John, On the motion of floating bodies. Ⅱ. Simple harmonic motions, Comm. Pure App. Math., 3 (1950), 45-101. doi: 10.1002/cpa.3160030106.
[13]	G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal Für Die Reine Und Angewandte Mathematik (Crelle's Journal), 71 (1869), 237-262. doi: 10.1515/crll.1870.71.237.
[14]	E. A. Kuznetsov and V. P. Ruban, Cherenkov interaction of vortices with a free surface, J. Exp. Theor. Phys., 88 (1999), 492-505. doi: 10.1134/1.558820.
[15]	S. D. Kelly and H. Xiong, Self-propulsion of a free hydrofoil with localized discrete vortex shedding: Analytical modeling and simulation, Theoretical and Computational Fluid Dynamics, 24, (2010), 45–50.
[16]	H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932.
[17]	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.
[18]	T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.
[19]	T. Miloh, Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.
[20]	T. Miloh, A note on impulsive sphere motion beneath a free-surface, J. Engg. Math., 41 (2001), 1-11. doi: 10.1023/A:1011895510151.
[21]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Second Edition, Springer-Verlag, 1999. doi: 10.1007/978-0-387-21792-5.
[22]	L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996.
[23]	E. P. Rood, Vorticity interactions with a free surface, in Fluid Vortices (ed. S. I. Green), Fluid Mechanics and Its Applications, 30, Springer, Dordrecht, (1995), 687–730.
[24]	A. Rouhi and J. Wright, Hamiltonian formulation for the motion of vortices in the presence of a free surface for ideal flow, Phys. Rev E, 48 (1993), 1850-1865. doi: 10.1103/PhysRevE.48.1850.
[25]	N. Salvesen, E. O. Tuck and O. Faltinsen, Ship motions and sea loads, Transactions of the Society of Naval Architects and Marine Engineers, 6 (1970), 1–30. Available from: www.marinecontrol.org/References/papers/SalvesenTuckFaltinsen1970.PDF
[26]	P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.
[27]	B. N. Shashikanth, Kirchhoff's equations of motion via a constrained Zakharov system, J. Geom. Mech., 8 (2016), 461-485. doi: 10.3934/jgm.2016016.
[28]	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.
[29]	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.
[30]	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.
[31]	P. Tallapragada and S. D. Kelly, Propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid, Regular and Chaotic Dynamics, 18 (2013), 21-32. doi: 10.1134/S1560354713010024.
[32]	F. Ursell, On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. App. Math., Ⅱ (1949), 218-231. doi: 10.1093/qjmam/2.2.218.
[33]	F. Ursell, Short surface waves due to an oscillating immersed body, Proc. R. Soc. Lond. A, 220 (1953), 90-103. doi: 10.1098/rspa.1953.0174.
[34]	E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993.
[35]	E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, St John's, Newfoundland, Canada, 1993,159–163. Available from: http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf.
[36]	J. Vankerschaver, E. Kanso and J. Marsden:, The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223.
[37]	J. V. Wehausen, The motion of floating bodies, Ann. Rev. Fluid Mech., 3 (1971), 237-268. doi: 10.1146/annurev.fl.03.010171.001321.
[38]	G. X. Wu and R. Eatock Taylor, The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Engg., 30 (2003), 387-400. doi: 10.1016/S0029-8018(02)00037-9.
[39]	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]	G. A. Athanassoulis, Analytical dynamics of wave body interaction, in Mathematical techniques for water waves (ed. B. N. Mandal), Advances in Fluid Mechanics, 8, Computational Mechanics Publications, Southampton, Boston (1997), 79–154. Available from: urlhttps://www.researchgate.net/publication/266301554_Analytical_dynamics_of_wave-body_interaction
[2]	G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, Cambridge, 1999.
[3]	M. Brøns, M. C. Thompson, T. Leweke and K. Hourigan, Vorticity generation and conservation for two-dimensional interfaces and boundaries, J. Fluid Mech., 758 (2014), 63-93.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[5]	A. Compelli, R. Ivanov and M. Todorov, Hamiltonian models for the propogation of irrotational surface gravity waves over a variable bottom, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170091, 15 pp. doi: 10.1098/rsta.2017.0091.
[6]	L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007.
[7]	C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys, Rev. Lett., 19 (1967), 1095-1097.
[8]	F. Gay-Balmaz, J. E. Marsden and T. S. Ratiu, Reduced variational formulations in free boundary continuum mechanics, J. Nonlinear Sci., 22 (2012), 463-497. doi: 10.1007/s00332-012-9143-4.
[9]	R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.
[10]	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.
[11]	F. John, On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57. doi: 10.1002/cpa.3160020102.
[12]	F. John, On the motion of floating bodies. Ⅱ. Simple harmonic motions, Comm. Pure App. Math., 3 (1950), 45-101. doi: 10.1002/cpa.3160030106.
[13]	G. Kirchhoff, Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal Für Die Reine Und Angewandte Mathematik (Crelle's Journal), 71 (1869), 237-262. doi: 10.1515/crll.1870.71.237.
[14]	E. A. Kuznetsov and V. P. Ruban, Cherenkov interaction of vortices with a free surface, J. Exp. Theor. Phys., 88 (1999), 492-505. doi: 10.1134/1.558820.
[15]	S. D. Kelly and H. Xiong, Self-propulsion of a free hydrofoil with localized discrete vortex shedding: Analytical modeling and simulation, Theoretical and Computational Fluid Dynamics, 24, (2010), 45–50.
[16]	H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932.
[17]	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.
[18]	T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.
[19]	T. Miloh, Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.
[20]	T. Miloh, A note on impulsive sphere motion beneath a free-surface, J. Engg. Math., 41 (2001), 1-11. doi: 10.1023/A:1011895510151.
[21]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Second Edition, Springer-Verlag, 1999. doi: 10.1007/978-0-387-21792-5.
[22]	L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996.
[23]	E. P. Rood, Vorticity interactions with a free surface, in Fluid Vortices (ed. S. I. Green), Fluid Mechanics and Its Applications, 30, Springer, Dordrecht, (1995), 687–730.
[24]	A. Rouhi and J. Wright, Hamiltonian formulation for the motion of vortices in the presence of a free surface for ideal flow, Phys. Rev E, 48 (1993), 1850-1865. doi: 10.1103/PhysRevE.48.1850.
[25]	N. Salvesen, E. O. Tuck and O. Faltinsen, Ship motions and sea loads, Transactions of the Society of Naval Architects and Marine Engineers, 6 (1970), 1–30. Available from: www.marinecontrol.org/References/papers/SalvesenTuckFaltinsen1970.PDF
[26]	P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.
[27]	B. N. Shashikanth, Kirchhoff's equations of motion via a constrained Zakharov system, J. Geom. Mech., 8 (2016), 461-485. doi: 10.3934/jgm.2016016.
[28]	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.
[29]	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.
[30]	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.
[31]	P. Tallapragada and S. D. Kelly, Propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid, Regular and Chaotic Dynamics, 18 (2013), 21-32. doi: 10.1134/S1560354713010024.
[32]	F. Ursell, On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. App. Math., Ⅱ (1949), 218-231. doi: 10.1093/qjmam/2.2.218.
[33]	F. Ursell, Short surface waves due to an oscillating immersed body, Proc. R. Soc. Lond. A, 220 (1953), 90-103. doi: 10.1098/rspa.1953.0174.
[34]	E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993.
[35]	E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, St John's, Newfoundland, Canada, 1993,159–163. Available from: http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf.
[36]	J. Vankerschaver, E. Kanso and J. Marsden:, The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223.
[37]	J. V. Wehausen, The motion of floating bodies, Ann. Rev. Fluid Mech., 3 (1971), 237-268. doi: 10.1146/annurev.fl.03.010171.001321.
[38]	G. X. Wu and R. Eatock Taylor, The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Engg., 30 (2003), 387-400. doi: 10.1016/S0029-8018(02)00037-9.
[39]	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.

Figure 1. Schematic perspective of a rigid body $ B $ beneath the free surface $ \Sigma_f $ of water. The bottom flat surface $ \mathcal{S} $ is shown by the dashed rectangle. Both $ \Sigma_f $ and $ \mathcal{S} $ extend to infinity in the $ x $ and $ y $ (horizontal) directions. In the text, the origin of the spatial frame $ xyz $ is located at the center of the disc $ C_R \subset \mathcal{S} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. A vertical slice of the setup in Figure 1, shown along with the body-fixed frame

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51
[2]	Frol Zapolsky. Quasi-states and the Poisson bracket on surfaces. Journal of Modern Dynamics, 2007, 1 (3) : 465-475. doi: 10.3934/jmd.2007.1.465
[3]	Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004
[4]	Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002
[5]	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
[6]	Boris Kolev. Poisson brackets in Hydrodynamics. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 555-574. doi: 10.3934/dcds.2007.19.555
[7]	Yacine Aït Amrane, Rafik Nasri, Ahmed Zeglaoui. Warped Poisson brackets on warped products. Journal of Geometric Mechanics, 2014, 6 (3) : 279-296. doi: 10.3934/jgm.2014.6.279
[8]	Christopher Griffin, James Fan. Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games. Journal of Dynamics and Games, 2022, 9 (2) : 165-189. doi: 10.3934/jdg.2022002
[9]	Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
[10]	Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37
[11]	Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61
[12]	Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic and Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063
[13]	Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345
[14]	Nancy López Reyes, Luis E. Benítez Babilonia. A discrete hierarchy of double bracket equations and a class of negative power series. Mathematical Control and Related Fields, 2017, 7 (1) : 41-52. doi: 10.3934/mcrf.2017003
[15]	Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 355-384. doi: 10.3934/jgm.2021024
[16]	Johannes Huebschmann. On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras. Journal of Geometric Mechanics, 2021, 13 (3) : 385-402. doi: 10.3934/jgm.2021009
[17]	Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052
[18]	Xianchao Wang, Jiaqi Zhu, Minghui Song, Wei Wu. Fourier method for reconstructing elastic body force from the coupled-wave field. Inverse Problems and Imaging, 2022, 16 (2) : 325-340. doi: 10.3934/ipi.2021052
[19]	Henrique Bursztyn, Alejandro Cabrera, Matias del Hoyo. Poisson double structures. Journal of Geometric Mechanics, 2022, 14 (2) : 151-178. doi: 10.3934/jgm.2021029
[20]	Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

2021 Impact Factor: 0.737

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 0.737

Tools

Figures and Tables

2021 Impact Factor: 0.737

American Institute of Mathematical Sciences

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors