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

Kinematic reduction and the Hamilton-Jacobi equation

1. 

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid, Spain

3. 

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid

4. 

Unidad asociada ULL-CSIC, Geometría diferencial y mecánica geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, 38071 La Laguna, Tenerife, Canary Islands, Spain

5. 

Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona, Spain

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, (1978). Google Scholar

[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. doi: 10.1088/0951-7715/23/8/006. Google Scholar

[3]

M. Barbero-Liñán and M. C. Muñoz Lecanda, Strict abnormal extremals in nonholonomic and kinematic control systems,, Special issue, 3 (2010), 1. Google Scholar

[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, (2005). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.3934/dcds.2009.24.213. Google Scholar

[9]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, IMA J. Math. Control. Inform., 21 (2004), 457. doi: 10.1093/imamci/21.4.457. Google Scholar

[10]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A: Math Theoret., 41 (2008), 175. Google Scholar

[11]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559. Google Scholar

[12]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3049752. Google Scholar

[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). doi: 10.1088/1751-8113/41/1/015205. Google Scholar

[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. Google Scholar

[15]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005). doi: 10.1088/0305-4470/38/24/R01. Google Scholar

[16]

M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North Holland Math. Series, 152 (1996). Google Scholar

[17]

P. Libermann, Lie algebroids and mechanics,, Arch. Math. (Brno), 32 (1996), 147. Google Scholar

[18]

K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids in Differential Geometry,", London Mathematical Society Lecture Note Series, 213 (2005). Google Scholar

[19]

J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 605. Google Scholar

[20]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259. Google Scholar

[21]

M. C. Muñoz-Lecanda and F. J. Yañiz-Fernández, Mechanical control systems and kinematic systems,, IEEE Trans. Automat. Control, 53 (2008), 1297. doi: 10.1109/TAC.2008.921004. Google Scholar

[22]

T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and Integrability,, Journal of Geometric Mechanics, 1 (2009), 461. Google Scholar

[23]

M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures,, in, 54 (2001), 217. Google Scholar

[24]

H. J. Sussmann, Orbits of families of vector fields and integrability of distributions,, Transactions of the American Mathematical Society, 180 (1973), 171. doi: 10.1090/S0002-9947-1973-0321133-2. Google Scholar

[25]

A. Weinstein, Lagrangian mechanics and groupoids,, in, 7 (1996), 207. Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", $2^{nd}$ edition, (1978). Google Scholar

[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. doi: 10.1088/0951-7715/23/8/006. Google Scholar

[3]

M. Barbero-Liñán and M. C. Muñoz Lecanda, Strict abnormal extremals in nonholonomic and kinematic control systems,, Special issue, 3 (2010), 1. Google Scholar

[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, (2005). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.3934/dcds.2009.24.213. Google Scholar

[9]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, IMA J. Math. Control. Inform., 21 (2004), 457. doi: 10.1093/imamci/21.4.457. Google Scholar

[10]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A: Math Theoret., 41 (2008), 175. Google Scholar

[11]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559. Google Scholar

[12]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3049752. Google Scholar

[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). doi: 10.1088/1751-8113/41/1/015205. Google Scholar

[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. Google Scholar

[15]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005). doi: 10.1088/0305-4470/38/24/R01. Google Scholar

[16]

M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North Holland Math. Series, 152 (1996). Google Scholar

[17]

P. Libermann, Lie algebroids and mechanics,, Arch. Math. (Brno), 32 (1996), 147. Google Scholar

[18]

K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids in Differential Geometry,", London Mathematical Society Lecture Note Series, 213 (2005). Google Scholar

[19]

J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 605. Google Scholar

[20]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259. Google Scholar

[21]

M. C. Muñoz-Lecanda and F. J. Yañiz-Fernández, Mechanical control systems and kinematic systems,, IEEE Trans. Automat. Control, 53 (2008), 1297. doi: 10.1109/TAC.2008.921004. Google Scholar

[22]

T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and Integrability,, Journal of Geometric Mechanics, 1 (2009), 461. Google Scholar

[23]

M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures,, in, 54 (2001), 217. Google Scholar

[24]

H. J. Sussmann, Orbits of families of vector fields and integrability of distributions,, Transactions of the American Mathematical Society, 180 (1973), 171. doi: 10.1090/S0002-9947-1973-0321133-2. Google Scholar

[25]

A. Weinstein, Lagrangian mechanics and groupoids,, in, 7 (1996), 207. Google Scholar

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