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

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

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.
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, (2005), 411.   Google Scholar

[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.  doi: 10.1142/S0219887806001235.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[4]

N. Borda, Sistemas Mecánicos Discretos con Vínculos de Orden 2,, Tesis de Maestría en Ciencias Físicas, (2011).   Google Scholar

[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.  doi: 10.1007/s13398-011-0053-3.  Google Scholar

[6]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006).  doi: 10.1063/1.2165797.  Google Scholar

[7]

H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2740470.  Google Scholar

[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.  doi: 10.1063/1.1763245.  Google Scholar

[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.  Google Scholar

[10]

N. G. Chetaev, On the Gauss principle,, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68.   Google Scholar

[11]

L. Colombo, D. Martín de Diego and M. Zuccalli, Higher-order discrete variational problems with constraints,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4820817.  Google Scholar

[12]

J. Cortés and S. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365.   Google Scholar

[13]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793).  doi: 10.1007/b84020.  Google Scholar

[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.  doi: 10.1017/S0305004100064501.  Google Scholar

[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, (1985).   Google Scholar

[16]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Geometric integrators and nonholonomic mechanics,, J. Math. Phys., 45 (2004), 1042.  doi: 10.1063/1.1644325.  Google Scholar

[17]

V. Dobronravov, The Fundamentals of the Mechanics of Nonholonomic Systems,, Vysshaya Shkola, (1970).   Google Scholar

[18]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[19]

S. Grillo, Higher order constraints Hamiltonian systems,, J. Math. Phys., 50 (2009).  doi: 10.1063/1.3194782.  Google Scholar

[20]

S. Grillo, Sistemas Noholónomos Generalizados,, Tesis de Doctorado en Matemática, (2007).   Google Scholar

[21]

S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, Int. J. Geom. Methods Mod. Phys., 7 (2010), 857.  doi: 10.1142/S0219887810004580.  Google Scholar

[22]

S. Grillo, J. E. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization,, J. Geom. Mech., 3 (2011), 145.  doi: 10.3934/jgm.2011.3.145.  Google Scholar

[23]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,, Second edition, (2006).   Google Scholar

[24]

O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.  doi: 10.1063/1.533411.  Google Scholar

[25]

C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353.   Google Scholar

[26]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar

[27]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Sci., 16 (2006), 283.  doi: 10.1007/s00332-005-0698-1.  Google Scholar

[28]

Yu. Neĭmark and N. Fufaev, Dynamics of Nonholonomic Systems,, Translations of Mathematical Monographs, (1972).   Google Scholar

[29]

G. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems,, Numer. Math., 113 (2009), 243.  doi: 10.1007/s00211-009-0245-3.  Google Scholar

[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.  doi: 10.1109/TAC.2005.852568.  Google Scholar

[31]

M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order,, AIP Conf. Proc., 1360 (2011), 164.  doi: 10.1063/1.3599143.  Google Scholar

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, (2005), 411.   Google Scholar

[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.  doi: 10.1142/S0219887806001235.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[4]

N. Borda, Sistemas Mecánicos Discretos con Vínculos de Orden 2,, Tesis de Maestría en Ciencias Físicas, (2011).   Google Scholar

[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.  doi: 10.1007/s13398-011-0053-3.  Google Scholar

[6]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006).  doi: 10.1063/1.2165797.  Google Scholar

[7]

H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2740470.  Google Scholar

[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.  doi: 10.1063/1.1763245.  Google Scholar

[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.  Google Scholar

[10]

N. G. Chetaev, On the Gauss principle,, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68.   Google Scholar

[11]

L. Colombo, D. Martín de Diego and M. Zuccalli, Higher-order discrete variational problems with constraints,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4820817.  Google Scholar

[12]

J. Cortés and S. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365.   Google Scholar

[13]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793).  doi: 10.1007/b84020.  Google Scholar

[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.  doi: 10.1017/S0305004100064501.  Google Scholar

[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, (1985).   Google Scholar

[16]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Geometric integrators and nonholonomic mechanics,, J. Math. Phys., 45 (2004), 1042.  doi: 10.1063/1.1644325.  Google Scholar

[17]

V. Dobronravov, The Fundamentals of the Mechanics of Nonholonomic Systems,, Vysshaya Shkola, (1970).   Google Scholar

[18]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[19]

S. Grillo, Higher order constraints Hamiltonian systems,, J. Math. Phys., 50 (2009).  doi: 10.1063/1.3194782.  Google Scholar

[20]

S. Grillo, Sistemas Noholónomos Generalizados,, Tesis de Doctorado en Matemática, (2007).   Google Scholar

[21]

S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, Int. J. Geom. Methods Mod. Phys., 7 (2010), 857.  doi: 10.1142/S0219887810004580.  Google Scholar

[22]

S. Grillo, J. E. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization,, J. Geom. Mech., 3 (2011), 145.  doi: 10.3934/jgm.2011.3.145.  Google Scholar

[23]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,, Second edition, (2006).   Google Scholar

[24]

O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.  doi: 10.1063/1.533411.  Google Scholar

[25]

C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353.   Google Scholar

[26]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar

[27]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Sci., 16 (2006), 283.  doi: 10.1007/s00332-005-0698-1.  Google Scholar

[28]

Yu. Neĭmark and N. Fufaev, Dynamics of Nonholonomic Systems,, Translations of Mathematical Monographs, (1972).   Google Scholar

[29]

G. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems,, Numer. Math., 113 (2009), 243.  doi: 10.1007/s00211-009-0245-3.  Google Scholar

[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.  doi: 10.1109/TAC.2005.852568.  Google Scholar

[31]

M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order,, AIP Conf. Proc., 1360 (2011), 164.  doi: 10.1063/1.3599143.  Google Scholar

[1]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[2]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[3]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[4]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[5]

David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002

[6]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[7]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[8]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[9]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[10]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[11]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[12]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202

[13]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[14]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[15]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[16]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[17]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[18]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[19]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

[20]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

[Back to Top]