# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017
##### References:
 [1] P. Appell, "Traite de Mechanique Rationelle,", Vol. 2, (1953). [2] P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595. [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. [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). [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. [6] T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws,, IEEE Trans. Automat. Control, 53 (2008), 339. [7] S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it,, in, (1956), 42. [8] S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem,, (in Russian) Math. Sbornik, 28 (1911), 303. [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). [10] M. J. Field, Equivariant dynamical systems,, Trans. Am. Math. Soc., 259 (1980), 185. [11] Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface,, (Russian) Izv. Akad. Nauk SSSR, 54 (1987), 67. [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. [13] Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh,, J. Phys. A, 43 (2010). [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, (2007). [16] J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493. [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. [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. [19] G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik,", Teubner, (1877). [20] H. Lamb, "Hydrodynamics,", $6^{th}$ edition, (1932). [21] N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica J. IFAC, 33 (1997), 331. [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, (1951). [23] L. Milne-Thomson, "Theoretical Hydrodynamics,", $5^{th}$ edition, (1968). [24] P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145,, Springer-Verlag, (2001). [25] S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid,, Reg. and Chaot. Dyn., 6 (2001), 33. [26] R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints,, in, (2000), 113. [27] J. Roenby and H. Aref, Chaos in body-vortex interactions,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871. [28] P. G. Saffman, "Vortex Dynamics,", Cambridge Monographs on Mechanics and Applied Mathematics, (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. 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. 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. 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. doi: 10.1134/S1560354710040143. [33] D. V. Zenkov, The geometry of the Routh problem,, J. Nonlinear Sci., 5 (1995), 503. doi: 10.1007/BF01209025.

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##### References:
 [1] P. Appell, "Traite de Mechanique Rationelle,", Vol. 2, (1953). [2] P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595. [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. [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). [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. [6] T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws,, IEEE Trans. Automat. Control, 53 (2008), 339. [7] S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it,, in, (1956), 42. [8] S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem,, (in Russian) Math. Sbornik, 28 (1911), 303. [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). [10] M. J. Field, Equivariant dynamical systems,, Trans. Am. Math. Soc., 259 (1980), 185. [11] Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface,, (Russian) Izv. Akad. Nauk SSSR, 54 (1987), 67. [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. [13] Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh,, J. Phys. A, 43 (2010). [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, (2007). [16] J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493. [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. [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. [19] G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik,", Teubner, (1877). [20] H. Lamb, "Hydrodynamics,", $6^{th}$ edition, (1932). [21] N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica J. IFAC, 33 (1997), 331. [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, (1951). [23] L. Milne-Thomson, "Theoretical Hydrodynamics,", $5^{th}$ edition, (1968). [24] P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145,, Springer-Verlag, (2001). [25] S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid,, Reg. and Chaot. Dyn., 6 (2001), 33. [26] R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints,, in, (2000), 113. [27] J. Roenby and H. Aref, Chaos in body-vortex interactions,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871. [28] P. G. Saffman, "Vortex Dynamics,", Cambridge Monographs on Mechanics and Applied Mathematics, (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. 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. 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. 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. doi: 10.1134/S1560354710040143. [33] D. V. Zenkov, The geometry of the Routh problem,, J. Nonlinear Sci., 5 (1995), 503. doi: 10.1007/BF01209025.
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