# American Institute of Mathematical Sciences

September  2012, 4(3): 207-237. doi: 10.3934/jgm.2012.4.207

## Kinematic reduction and the Hamilton-Jacobi equation

Received  October 2011 Revised  March 2012 Published  October 2012

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.
Citation: María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," $2^{nd}$ edition, Benjamin, New York, 1978. [2] P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics: Hamilton-Jacobi equation and applications, Nonlinearity, 23 (2010), 1887-1918. doi: 10.1088/0951-7715/23/8/006. [3] M. Barbero-Liñán and M. C. Muñoz Lecanda, Strict abnormal extremals in nonholonomic and kinematic control systems, Special issue "Nonholonomic constraints in Mechanics and Optimal Control Theory" in Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 1-17. [4] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, Springer Verlag, New York, 2005. [5] J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Roman-Roy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417-1458. [6] J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Roman-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431-454. [7] J. Cortés, M. de León, J. C. Marrero, D. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558. [8] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213. [9] J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA J. Math. Control. Inform., 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457. [10] K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math Theoret., 41 (2008), 175-204. [11] K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575. [12] J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 17 pp. 013520. doi: 10.1063/1.3049752. [13] D. Iglesias, M. de Le\ón and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A: Math. Theor., 41 (2008), 14 pp. 015205. doi: 10.1088/1751-8113/41/1/015205. [14] M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi theory. Applications to nonholonomic mechanics, Journal of Geometric Mechanics, 2 (2010), 159-198. [15] M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01. [16] M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North Holland Math. Series, 152, Amsterdam, 1996. [17] P. Libermann, Lie algebroids and mechanics, Arch. Math. (Brno), 32 (1996), 147-162. [18] K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids in Differential Geometry," London Mathematical Society Lecture Note Series, 213, 2005. [19] J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 605-622. [20] E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259. [21] M. C. Muñoz-Lecanda and F. J. Yañiz-Fernández, Mechanical control systems and kinematic systems, IEEE Trans. Automat. Control, 53 (2008), 1297-1302. doi: 10.1109/TAC.2008.921004. [22] T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and Integrability, Journal of Geometric Mechanics, 1 (2009), 461-481. [23] M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures, in "Proc. Workshop on Lie Algebroids and Related Topics in Differential Geometry" (Warsaw), Warsaw: Banach Center Publications, 54 (2001), 217-233. [24] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Transactions of the American Mathematical Society, 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2. [25] A. Weinstein, Lagrangian mechanics and groupoids, in "Proceedings of Mechanics Day" (Waterloo, ON, 1992), Fields Institute Communications, Amer. Math. Soc., 7 (1996), 207-231.

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##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," $2^{nd}$ edition, Benjamin, New York, 1978. [2] P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics: Hamilton-Jacobi equation and applications, Nonlinearity, 23 (2010), 1887-1918. doi: 10.1088/0951-7715/23/8/006. [3] M. Barbero-Liñán and M. C. Muñoz Lecanda, Strict abnormal extremals in nonholonomic and kinematic control systems, Special issue "Nonholonomic constraints in Mechanics and Optimal Control Theory" in Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 1-17. [4] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, Springer Verlag, New York, 2005. [5] J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Roman-Roy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417-1458. [6] J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Roman-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431-454. [7] J. Cortés, M. de León, J. C. Marrero, D. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558. [8] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213. [9] J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA J. Math. Control. Inform., 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457. [10] K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math Theoret., 41 (2008), 175-204. [11] K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575. [12] J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 17 pp. 013520. doi: 10.1063/1.3049752. [13] D. Iglesias, M. de Le\ón and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A: Math. Theor., 41 (2008), 14 pp. 015205. doi: 10.1088/1751-8113/41/1/015205. [14] M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi theory. Applications to nonholonomic mechanics, Journal of Geometric Mechanics, 2 (2010), 159-198. [15] M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01. [16] M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North Holland Math. Series, 152, Amsterdam, 1996. [17] P. Libermann, Lie algebroids and mechanics, Arch. Math. (Brno), 32 (1996), 147-162. [18] K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids in Differential Geometry," London Mathematical Society Lecture Note Series, 213, 2005. [19] J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 605-622. [20] E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259. [21] M. C. Muñoz-Lecanda and F. J. Yañiz-Fernández, Mechanical control systems and kinematic systems, IEEE Trans. Automat. Control, 53 (2008), 1297-1302. doi: 10.1109/TAC.2008.921004. [22] T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and Integrability, Journal of Geometric Mechanics, 1 (2009), 461-481. [23] M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures, in "Proc. Workshop on Lie Algebroids and Related Topics in Differential Geometry" (Warsaw), Warsaw: Banach Center Publications, 54 (2001), 217-233. [24] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Transactions of the American Mathematical Society, 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2. [25] A. Weinstein, Lagrangian mechanics and groupoids, in "Proceedings of Mechanics Day" (Waterloo, ON, 1992), Fields Institute Communications, Amer. Math. Soc., 7 (1996), 207-231.
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