# American Institute of Mathematical Sciences

June  2009, 1(2): 223-266. doi: 10.3934/jgm.2009.1.223

## The geometry and dynamics of interacting rigid bodies and point vortices

 1 Control and Dynamical Systems, California Institute of Technology MC 107-81, Pasadena, CA 91125, United States 2 Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, United States 3 Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States

Received  November 2008 Revised  March 2009 Published  July 2009

We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabeling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics.
Citation: Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223
 [1] A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100 [2] Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13 [3] Cesare Tronci. Hybrid models for perfect complex fluids with multipolar interactions. Journal of Geometric Mechanics, 2012, 4 (3) : 333-363. doi: 10.3934/jgm.2012.4.333 [4] Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439 [5] Takashi Suzuki. Brownian point vortices and dd-model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161 [6] A. V. Borisov, I. S. Mamaev, S. M. Ramodanov. Dynamics of a circular cylinder interacting with point vortices. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 35-50. doi: 10.3934/dcdsb.2005.5.35 [7] Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927 [8] Colm Connaughton, John R. Ockendon. Interactions of point vortices in the Zabusky-McWilliams model with a background flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1795-1807. doi: 10.3934/dcdsb.2012.17.1795 [9] Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023 [10] C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7 [11] L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395 [12] Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223 [13] Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 [14] Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399 [15] Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193 [16] Olof Heden. A survey of perfect codes. Advances in Mathematics of Communications, 2008, 2 (2) : 223-247. doi: 10.3934/amc.2008.2.223 [17] Luciano Panek, Jerry Anderson Pinheiro, Marcelo Muniz Alves, Marcelo Firer. On perfect poset codes. Advances in Mathematics of Communications, 2020, 14 (3) : 477-489. doi: 10.3934/amc.2020061 [18] Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov. Perfect retroreflectors and billiard dynamics. Journal of Modern Dynamics, 2011, 5 (1) : 33-48. doi: 10.3934/jmd.2011.5.33 [19] Marcela Mejía, J. Urías. An asymptotically perfect pseudorandom generator. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 115-126. doi: 10.3934/dcds.2001.7.115 [20] Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349

2020 Impact Factor: 0.857