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A special tribute to Professor Pedro L. García
Discrete second order constrained Lagrangian systems: First results
1. | Depto. de Matemática, Facultad de Ciencias Exactas, UNLP, Instituto Balseiro, UNCu - CNEA - CONICET, 50 y 115, La Plata, Buenos Aires, 1900, Argentina |
2. | Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A., Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP |
3. | Instituto Balseiro, UNCu - CNEA - CONICET, Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, Argentina |
References:
[1] |
R. Benito and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms, in Differential Geometry and its Applications, Matfyzpress, Prague, 2005, 411-419. |
[2] |
R. Benito, M. de León and D. Martín de Diego, Higher-order discrete Lagrangian mechanics, Int. J. Geom. Methods Mod. Phys., 3 (2006), 421-436.
doi: 10.1142/S0219887806001235. |
[3] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. |
[4] |
N. Borda, Sistemas Mecánicos Discretos con Vínculos de Orden 2, Tesis de Maestría en Ciencias Físicas, Instituto Balseiro, S. C. de Bariloche, 2011. |
[5] |
C. M. Campos, H. Cendra, V. Díaz and D. Martín de Diego, Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 225-234.
doi: 10.1007/s13398-011-0053-3. |
[6] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.
doi: 10.1063/1.2165797. |
[7] |
H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.
doi: 10.1063/1.2740470. |
[8] |
H. Cendra, A. Ibort, M. de León and D. Martín de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801.
doi: 10.1063/1.1763245. |
[9] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001).
doi: 10.1090/memo/0722. |
[10] |
N. G. Chetaev, On the Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71. |
[11] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Higher-order discrete variational problems with constraints, J. Math. Phys., 54 (2013), 093507.
doi: 10.1063/1.4820817. |
[12] |
J. Cortés and S. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. |
[13] |
J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b84020. |
[14] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587.
doi: 10.1017/S0305004100064501. |
[15] |
M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives, North-Holland Mathematics Studies, 112, Notes on Pure Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1985. |
[16] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys., 45 (2004), 1042-1064.
doi: 10.1063/1.1644325. |
[17] |
V. Dobronravov, The Fundamentals of the Mechanics of Nonholonomic Systems, Vysshaya Shkola, 1970. |
[18] |
J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111.
doi: 10.3934/jgm.2010.2.69. |
[19] |
S. Grillo, Higher order constraints Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.
doi: 10.1063/1.3194782. |
[20] |
S. Grillo, Sistemas Noholónomos Generalizados, Tesis de Doctorado en Matemática, Universidad Nacional del Sur, Bahía Blanca, 2007. |
[21] |
S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 857-886.
doi: 10.1142/S0219887810004580. |
[22] |
S. Grillo, J. E. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, J. Geom. Mech., 3 (2011), 145-196.
doi: 10.3934/jgm.2011.3.145. |
[23] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. |
[24] |
O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324.
doi: 10.1063/1.533411. |
[25] |
C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. |
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[27] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[28] |
Yu. Neĭmark and N. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, AMS, Providence, 1972. |
[29] |
G. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.
doi: 10.1007/s00211-009-0245-3. |
[30] |
A. Shiriaev, J. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach, IEEE Trans. Automat. Control, 50 (2005), 1164-1176.
doi: 10.1109/TAC.2005.852568. |
[31] |
M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order, AIP Conf. Proc., 1360 (2011), 164-169.
doi: 10.1063/1.3599143. |
show all references
References:
[1] |
R. Benito and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms, in Differential Geometry and its Applications, Matfyzpress, Prague, 2005, 411-419. |
[2] |
R. Benito, M. de León and D. Martín de Diego, Higher-order discrete Lagrangian mechanics, Int. J. Geom. Methods Mod. Phys., 3 (2006), 421-436.
doi: 10.1142/S0219887806001235. |
[3] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. |
[4] |
N. Borda, Sistemas Mecánicos Discretos con Vínculos de Orden 2, Tesis de Maestría en Ciencias Físicas, Instituto Balseiro, S. C. de Bariloche, 2011. |
[5] |
C. M. Campos, H. Cendra, V. Díaz and D. Martín de Diego, Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 225-234.
doi: 10.1007/s13398-011-0053-3. |
[6] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.
doi: 10.1063/1.2165797. |
[7] |
H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.
doi: 10.1063/1.2740470. |
[8] |
H. Cendra, A. Ibort, M. de León and D. Martín de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801.
doi: 10.1063/1.1763245. |
[9] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001).
doi: 10.1090/memo/0722. |
[10] |
N. G. Chetaev, On the Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71. |
[11] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Higher-order discrete variational problems with constraints, J. Math. Phys., 54 (2013), 093507.
doi: 10.1063/1.4820817. |
[12] |
J. Cortés and S. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. |
[13] |
J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b84020. |
[14] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587.
doi: 10.1017/S0305004100064501. |
[15] |
M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives, North-Holland Mathematics Studies, 112, Notes on Pure Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1985. |
[16] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys., 45 (2004), 1042-1064.
doi: 10.1063/1.1644325. |
[17] |
V. Dobronravov, The Fundamentals of the Mechanics of Nonholonomic Systems, Vysshaya Shkola, 1970. |
[18] |
J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111.
doi: 10.3934/jgm.2010.2.69. |
[19] |
S. Grillo, Higher order constraints Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.
doi: 10.1063/1.3194782. |
[20] |
S. Grillo, Sistemas Noholónomos Generalizados, Tesis de Doctorado en Matemática, Universidad Nacional del Sur, Bahía Blanca, 2007. |
[21] |
S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 857-886.
doi: 10.1142/S0219887810004580. |
[22] |
S. Grillo, J. E. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, J. Geom. Mech., 3 (2011), 145-196.
doi: 10.3934/jgm.2011.3.145. |
[23] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. |
[24] |
O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324.
doi: 10.1063/1.533411. |
[25] |
C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. |
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[27] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[28] |
Yu. Neĭmark and N. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, AMS, Providence, 1972. |
[29] |
G. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.
doi: 10.1007/s00211-009-0245-3. |
[30] |
A. Shiriaev, J. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach, IEEE Trans. Automat. Control, 50 (2005), 1164-1176.
doi: 10.1109/TAC.2005.852568. |
[31] |
M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order, AIP Conf. Proc., 1360 (2011), 164-169.
doi: 10.1063/1.3599143. |
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