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Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes
1. | Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom |
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, American Mathematical Society, 2000. |
[2] |
R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 3rd edition, Spinger, 2009. |
[3] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 316-361.
doi: 10.5802/aif.233. |
[4] |
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, vol. 24 of Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998. |
[5] |
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999. |
[6] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp.
doi: 10.1090/memo/0722. |
[7] |
R. L. Fernandes and I. Struchiner, Lie algebroids and classification problems in geometry, São Paulo J. Math. Sci., 2 (2008), 263-283. |
[8] |
E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, 240 (2011), 1724-1760.
doi: 10.1016/j.physd.2011.07.011. |
[9] |
F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176-275.
doi: 10.1016/j.aam.2008.06.002. |
[10] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Series in Computational Mathematics, Springer Verlag, 2002. |
[11] |
H. O. Jacobs, T. S. Ratiu and M. Desbrun, On the coupling between an ideal fluid and immersed particles, Phys. D, 265 (2013), 40-56. arXiv:1208.6561v1
doi: 10.1016/j.physd.2013.09.004. |
[12] |
E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science, 15 (2005), 255-289.
doi: 10.1007/s00332-004-0650-9. |
[13] |
S. D. Kelly, The Mechanics and Control of Robotic Locomotion with Applications to Aquatic Vehicles, PhD thesis, California Institute of Technology, 1998. |
[14] |
S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, John Wiley & Sons, 1963. |
[15] |
I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. |
[16] |
H. Lamb, Hydrodynamics, Reprint of the 1932 Cambridge University Press edition, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. |
[17] |
T. Lee, M. Leok and N. H. McClamroch, Computational geometric optimal control of rigid bodies, Communications in Information and Systems, 8 (2008), 445-472.
doi: 10.4310/CIS.2008.v8.n4.a5. |
[18] |
D. Lewis, J. E. Marsden, R. Montgomery and T. S. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Phys. D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[19] |
J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348.
doi: 10.1088/0951-7715/19/6/006. |
[20] |
J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Corrected reprint of the 1983 original. Dover Publications, Inc., New York, 1994. |
[21] |
J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.
doi: 10.1007/BF00914351. |
[22] |
J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164. |
[23] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[24] |
J. E. Radford, Symmetry, Reduction and Swimming in a Perfect Fluid, PhD thesis, California Institute of Technology, 2003. |
[25] |
G. Schwarz, Hodge Decomposition-A Method for Solving Boundary Value Problems, vol. 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995. |
[26] |
A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585.
doi: 10.1017/S002211208900025X. |
[27] |
M. Troyanov, On the Hodge decomposition in $\mathbb{R}^{N}$, Mosc. Math. J., 9 (2009), 899-926, 936. |
[28] |
J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, Journal of Geometric Mechanics, 1 (2009), 223-266.
doi: 10.3934/jgm.2009.1.223. |
[29] |
J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 606-629.
doi: 10.1134/S1560354710040143. |
[30] |
A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, American Mathematical Society, 2000. |
[2] |
R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 3rd edition, Spinger, 2009. |
[3] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 316-361.
doi: 10.5802/aif.233. |
[4] |
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, vol. 24 of Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998. |
[5] |
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999. |
[6] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp.
doi: 10.1090/memo/0722. |
[7] |
R. L. Fernandes and I. Struchiner, Lie algebroids and classification problems in geometry, São Paulo J. Math. Sci., 2 (2008), 263-283. |
[8] |
E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, 240 (2011), 1724-1760.
doi: 10.1016/j.physd.2011.07.011. |
[9] |
F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176-275.
doi: 10.1016/j.aam.2008.06.002. |
[10] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Series in Computational Mathematics, Springer Verlag, 2002. |
[11] |
H. O. Jacobs, T. S. Ratiu and M. Desbrun, On the coupling between an ideal fluid and immersed particles, Phys. D, 265 (2013), 40-56. arXiv:1208.6561v1
doi: 10.1016/j.physd.2013.09.004. |
[12] |
E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science, 15 (2005), 255-289.
doi: 10.1007/s00332-004-0650-9. |
[13] |
S. D. Kelly, The Mechanics and Control of Robotic Locomotion with Applications to Aquatic Vehicles, PhD thesis, California Institute of Technology, 1998. |
[14] |
S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, John Wiley & Sons, 1963. |
[15] |
I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. |
[16] |
H. Lamb, Hydrodynamics, Reprint of the 1932 Cambridge University Press edition, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. |
[17] |
T. Lee, M. Leok and N. H. McClamroch, Computational geometric optimal control of rigid bodies, Communications in Information and Systems, 8 (2008), 445-472.
doi: 10.4310/CIS.2008.v8.n4.a5. |
[18] |
D. Lewis, J. E. Marsden, R. Montgomery and T. S. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Phys. D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[19] |
J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348.
doi: 10.1088/0951-7715/19/6/006. |
[20] |
J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Corrected reprint of the 1983 original. Dover Publications, Inc., New York, 1994. |
[21] |
J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.
doi: 10.1007/BF00914351. |
[22] |
J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164. |
[23] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[24] |
J. E. Radford, Symmetry, Reduction and Swimming in a Perfect Fluid, PhD thesis, California Institute of Technology, 2003. |
[25] |
G. Schwarz, Hodge Decomposition-A Method for Solving Boundary Value Problems, vol. 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995. |
[26] |
A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585.
doi: 10.1017/S002211208900025X. |
[27] |
M. Troyanov, On the Hodge decomposition in $\mathbb{R}^{N}$, Mosc. Math. J., 9 (2009), 899-926, 936. |
[28] |
J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, Journal of Geometric Mechanics, 1 (2009), 223-266.
doi: 10.3934/jgm.2009.1.223. |
[29] |
J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 606-629.
doi: 10.1134/S1560354710040143. |
[30] |
A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996. |
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