# American Institute of Mathematical Sciences

September  2013, 33(9): 4017-4040. doi: 10.3934/dcds.2013.33.4017

## The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

 1 Department de Matemática Aplicada I, Universitat Politecnica de Catalunya, Barcelona, E-08028, Spain 2 Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, Rio Hondo 1, Mexico City, 01000, Mexico 3 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  January 2012 Revised  December 2012 Published  March 2013

We consider the motion of a planar rigid body in a potential two-dimensional flow with a circulation and subject to a certain nonholonomic constraint. This model can be related to the design of underwater vehicles.
The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.
Citation: Yuri N. Fedorov, Luis C. García-Naranjo, Joris Vankerschaver. The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017
##### References:
 [1] P. Appell, "Traite de Mechanique Rationelle," Vol. 2, Gauthier-Villars, Paris, 1953. [2] P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. [3] A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. and Chaot. Dyn., 8 (2003), 449-462. [4] A. V. Borisov and I. S. Mamaev, On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 16 (2006), 013118, 7 pp. [5] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlin. Sci., 21 (2011), 325-385. [6] T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws, IEEE Trans. Automat. Control, 53 (2008), 339-349. [7] S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it, in "The Selected Works on Wing Theory of Sergei A. Chaplygin," Translated from the 1933 Russian original by M. A. Garbell, (1956), 42-72. [8] S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem, (in Russian) Math. Sbornik, 28 (1911), 303-314; English translation in Reg. Chaot. Dyn., 13 (2008), 369-376. [9] F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 051, 12 pp. [10] M. J. Field, Equivariant dynamical systems, Trans. Am. Math. Soc., 259 (1980), 185-205. [11] Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface, (Russian) Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 54 (1987), 67-75; Eng. transl. in Mechanics of Solids, 22 (1988), 65-73. [12] Yu. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. [13] Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A, 43 (2010), 434013, 18 pp. [14] L. C. García-Naranjo and J. Vankerschaver, Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation,, preprint, (). [15] I. S. Gradshteyn and I. M. Ryzhik, "Tables of Integrals, Series, and Products," $7^{th}$ edition, Elsevier/Academic Press, Amsterdam, 2007. [16] J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. [17] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid J. Nonlinear Sci., 15 (2005), 255-289. [18] S. Kelly and R. Hukkeri, Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift, IEEE Transactions on Robotics, 22 (2006), 1254-1264. [19] G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik," Teubner, Leipzig, 1877. [20] H. Lamb, "Hydrodynamics," $6^{th}$ edition, Dover, NY, 1932. [21] N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica J. IFAC, 33 (1997), 331-346. [22] T. Levi-Civita and U. Amaldi, "Lezioni di Meccanica Razionale. Vol. 2. Dinamica dei Sistemi con un Numero Finito di Gradi di Liberta," (Italian) Nuova ed. N. Zanichelli, Bologna, 1951, 1952. [23] L. Milne-Thomson, "Theoretical Hydrodynamics," $5^{th}$ edition, revised and enlarged edition, MacMillan and Co. Ltd., London, 1968. [24] P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001. [25] S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid, Reg. and Chaot. Dyn., 6 (2001), 33-38. [26] R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints, in "Proceedings of the First International Symposium on Impact and Friction of Solids, Structures, and Intelligent Machines" (ed. A. Guran), World Scientific, Singapore, (2000), 113-116. [27] J. Roenby and H. Aref, Chaos in body-vortex interactions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871-1891. [28] P. G. Saffman, "Vortex Dynamics," Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. [29] B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227. doi: 10.1063/1.1445183. [30] B. N. Shashikanth, A. Sheshmani, S. D. Kelly and W. Mingjun, Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings, J. Math. Fluid. Mech., 12 (2010), 335-353. doi: 10.1007/s00021-008-0291-0. [31] J. Vankerschaver, E. Kanso and J. E. 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. [32] J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 609-629. doi: 10.1134/S1560354710040143. [33] D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.

show all references

##### References:
 [1] P. Appell, "Traite de Mechanique Rationelle," Vol. 2, Gauthier-Villars, Paris, 1953. [2] P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. [3] A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. and Chaot. Dyn., 8 (2003), 449-462. [4] A. V. Borisov and I. S. Mamaev, On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 16 (2006), 013118, 7 pp. [5] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlin. Sci., 21 (2011), 325-385. [6] T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws, IEEE Trans. Automat. Control, 53 (2008), 339-349. [7] S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it, in "The Selected Works on Wing Theory of Sergei A. Chaplygin," Translated from the 1933 Russian original by M. A. Garbell, (1956), 42-72. [8] S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem, (in Russian) Math. Sbornik, 28 (1911), 303-314; English translation in Reg. Chaot. Dyn., 13 (2008), 369-376. [9] F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 051, 12 pp. [10] M. J. Field, Equivariant dynamical systems, Trans. Am. Math. Soc., 259 (1980), 185-205. [11] Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface, (Russian) Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 54 (1987), 67-75; Eng. transl. in Mechanics of Solids, 22 (1988), 65-73. [12] Yu. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. [13] Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A, 43 (2010), 434013, 18 pp. [14] L. C. García-Naranjo and J. Vankerschaver, Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation,, preprint, (). [15] I. S. Gradshteyn and I. M. Ryzhik, "Tables of Integrals, Series, and Products," $7^{th}$ edition, Elsevier/Academic Press, Amsterdam, 2007. [16] J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. [17] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid J. Nonlinear Sci., 15 (2005), 255-289. [18] S. Kelly and R. Hukkeri, Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift, IEEE Transactions on Robotics, 22 (2006), 1254-1264. [19] G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik," Teubner, Leipzig, 1877. [20] H. Lamb, "Hydrodynamics," $6^{th}$ edition, Dover, NY, 1932. [21] N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica J. IFAC, 33 (1997), 331-346. [22] T. Levi-Civita and U. Amaldi, "Lezioni di Meccanica Razionale. Vol. 2. Dinamica dei Sistemi con un Numero Finito di Gradi di Liberta," (Italian) Nuova ed. N. Zanichelli, Bologna, 1951, 1952. [23] L. Milne-Thomson, "Theoretical Hydrodynamics," $5^{th}$ edition, revised and enlarged edition, MacMillan and Co. Ltd., London, 1968. [24] P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001. [25] S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid, Reg. and Chaot. Dyn., 6 (2001), 33-38. [26] R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints, in "Proceedings of the First International Symposium on Impact and Friction of Solids, Structures, and Intelligent Machines" (ed. A. Guran), World Scientific, Singapore, (2000), 113-116. [27] J. Roenby and H. Aref, Chaos in body-vortex interactions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871-1891. [28] P. G. Saffman, "Vortex Dynamics," Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. [29] B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227. doi: 10.1063/1.1445183. [30] B. N. Shashikanth, A. Sheshmani, S. D. Kelly and W. Mingjun, Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings, J. Math. Fluid. Mech., 12 (2010), 335-353. doi: 10.1007/s00021-008-0291-0. [31] J. Vankerschaver, E. Kanso and J. E. 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. [32] J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 609-629. doi: 10.1134/S1560354710040143. [33] D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.
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