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December  2013, 5(4): 381-397. doi: 10.3934/jgm.2013.5.381

Discrete second order constrained Lagrangian systems: First results

Nicolás Borda 1, , Javier Fernández 2, and  Sergio Grillo 3,

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

Received  March 2013 Revised  November 2013 Published  December 2013

We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of two mechanical systems: a particle moving in the plane with prescribed signed curvature, and the inertia wheel pendulum with a Lyapunov constraint. In addition, we prove a local existence and uniqueness result for trajectories of DSOCSs. As a first comparison of the underlying geometric structures, we study the symplectic behavior of both SOCSs and DSOCSs.
Keywords: Geometric mechanics, nonholonomic mechanics, second order constraints., discrete mechanical systems.
Mathematics Subject Classification: Primary: 70F25, 70G75, 70H45; Secondary: 70G4.
Citation: Nicolás Borda, Javier Fernández, Sergio Grillo. Discrete second order constrained Lagrangian systems: First results. Journal of Geometric Mechanics, 2013, 5 (4) : 381-397. doi: 10.3934/jgm.2013.5.381
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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