December  2010, 2(4): 343-374. doi: 10.3934/jgm.2010.2.343

Variational integrators for discrete Lagrange problems

1. 

Department of Mathematics, University of Salamanca, Salamanca 37008, Spain

2. 

Department of Applied Mathematics, University of Salamanca, Salamanca 37008, Spain

3. 

CINAMIL, Academia Militar, Amadora 2720-113, Portugal

Received  August 2010 Revised  December 2010 Published  January 2011

A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1-jets of the discrete fibred manifold that configures the system. After defining the concepts of admissible section and infinitesimal admissible variation, the objective of these problems is to find admissible sections that are critical for the Lagrangian of the system with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). Next, we define the concept of Cartan 1-form, establish a Noether theory for symmetries and introduce a notion of "constrained variational integrator" that we characterize through a Cartan equation ensuring its symplecticity. Under a certain regularity condition of the problem of Lagrange, we prove the existence and uniqueness of this kind of integrators in the neighborhood of a critical section, showing then that such integrators can be constructed from a generating function of the second class in the sense of symplectic geometry. Finally, the whole theory is illustrated with three elementary examples.
Citation: Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343
References:
[1]

V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, "Dynamical Systems III,", Encyclopaedia of Mathematical Sciences, 3 (1988).   Google Scholar

[2]

R. Benito and D. Martín de Diego, Discrete vakonomic mechanics,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.2008214.  Google Scholar

[3]

A. M. Bloch, "Nonholonomic Mechanics and Control,'', Interdisciplinary Applied Mathematics, 24 (2003).   Google Scholar

[4]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints,, J. Geom. Phys., 18 (1996), 295.  doi: 10.1016/0393-0440(95)00016-X.  Google Scholar

[5]

J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energy-momentum integrators,, preprint, ().   Google Scholar

[6]

J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics,, Appl. Math. Comput., 177 (2006), 226.  doi: 10.1016/j.amc.2005.11.002.  Google Scholar

[7]

J. Cortés, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'', Lect. Notes in Math. \textbf{1793}, 1793 (2002).   Google Scholar

[8]

P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems,, J. Geom. Phys., 56 (2006), 571.  doi: 10.1016/j.geomphys.2005.04.002.  Google Scholar

[9]

P. L. García and C. Rodrigo, Cartan forms and second variation for constrained variational problems,, Proceedings of the VII International Conference on Geometry, (2006), 140.   Google Scholar

[10]

H. Goldstein, "Classical Mechanics,'', Addison-Wesley Series in Physics, (1980).   Google Scholar

[11]

X. Gràcia, J. Marín Solano and M. C. Muñoz Lecanda, Some geometric aspects of variational calculus in constrained systems,, Rep. Math. Phys., 51 (2003), 127.  doi: 10.1016/S0034-4877(03)80006-X.  Google Scholar

[12]

V. M. Guibout and A. Bloch, Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems,, e-print ccsd-00002863, (): 1.   Google Scholar

[13]

L. Hsu, Calculus of variations via the Griffiths formalism,, J. Diff. Geom., 36 (1992), 551.   Google Scholar

[14]

T. D. Lee, Can time be a discrete dynamical variable?,, Phys. Lett. B, 122 (1983).  doi: 10.1016/0370-2693(83)90687-1.  Google Scholar

[15]

M. de León, D. Martín de Diego and A. Santamaría Merino, Geometric integrators and nonholonomic mechanics,, J. Math. Phys., 45 (2004).   Google Scholar

[16]

M. de León, D. Martín de Diego and A. Santamaría Merino, Discrete variational integrators and optimal control theory,, Advances in Computational Mathematics, 26 (2006), 251.   Google Scholar

[17]

M. de León, J. C. Marrero and D. Martín de Diego, Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach,, J. Geom. Phys., 35 (2000), 126.  doi: 10.1016/S0393-0440(00)00004-8.  Google Scholar

[18]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs,, Comm. in Math. Phys., 199 (1998), 351.   Google Scholar

[19]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 317.  doi: 10.1017/S096249290100006X.  Google Scholar

[20]

S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control,, J. Geom. Phys., 38 (2001), 343.  doi: 10.1016/S0393-0440(00)00069-3.  Google Scholar

[21]

P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417.   Google Scholar

[22]

J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories,, Rep. Math. Phys., 56 (2005), 387.  doi: 10.1016/S0034-4877(05)80093-X.  Google Scholar

[23]

J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids,, J. Geom. Phys., 57 (2007), 665.  doi: 10.1016/j.geomphys.2006.05.006.  Google Scholar

[24]

M. West, "Variational Integrators,'', Ph.D. Thesis, (2004).   Google Scholar

show all references

References:
[1]

V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, "Dynamical Systems III,", Encyclopaedia of Mathematical Sciences, 3 (1988).   Google Scholar

[2]

R. Benito and D. Martín de Diego, Discrete vakonomic mechanics,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.2008214.  Google Scholar

[3]

A. M. Bloch, "Nonholonomic Mechanics and Control,'', Interdisciplinary Applied Mathematics, 24 (2003).   Google Scholar

[4]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints,, J. Geom. Phys., 18 (1996), 295.  doi: 10.1016/0393-0440(95)00016-X.  Google Scholar

[5]

J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energy-momentum integrators,, preprint, ().   Google Scholar

[6]

J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics,, Appl. Math. Comput., 177 (2006), 226.  doi: 10.1016/j.amc.2005.11.002.  Google Scholar

[7]

J. Cortés, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'', Lect. Notes in Math. \textbf{1793}, 1793 (2002).   Google Scholar

[8]

P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems,, J. Geom. Phys., 56 (2006), 571.  doi: 10.1016/j.geomphys.2005.04.002.  Google Scholar

[9]

P. L. García and C. Rodrigo, Cartan forms and second variation for constrained variational problems,, Proceedings of the VII International Conference on Geometry, (2006), 140.   Google Scholar

[10]

H. Goldstein, "Classical Mechanics,'', Addison-Wesley Series in Physics, (1980).   Google Scholar

[11]

X. Gràcia, J. Marín Solano and M. C. Muñoz Lecanda, Some geometric aspects of variational calculus in constrained systems,, Rep. Math. Phys., 51 (2003), 127.  doi: 10.1016/S0034-4877(03)80006-X.  Google Scholar

[12]

V. M. Guibout and A. Bloch, Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems,, e-print ccsd-00002863, (): 1.   Google Scholar

[13]

L. Hsu, Calculus of variations via the Griffiths formalism,, J. Diff. Geom., 36 (1992), 551.   Google Scholar

[14]

T. D. Lee, Can time be a discrete dynamical variable?,, Phys. Lett. B, 122 (1983).  doi: 10.1016/0370-2693(83)90687-1.  Google Scholar

[15]

M. de León, D. Martín de Diego and A. Santamaría Merino, Geometric integrators and nonholonomic mechanics,, J. Math. Phys., 45 (2004).   Google Scholar

[16]

M. de León, D. Martín de Diego and A. Santamaría Merino, Discrete variational integrators and optimal control theory,, Advances in Computational Mathematics, 26 (2006), 251.   Google Scholar

[17]

M. de León, J. C. Marrero and D. Martín de Diego, Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach,, J. Geom. Phys., 35 (2000), 126.  doi: 10.1016/S0393-0440(00)00004-8.  Google Scholar

[18]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs,, Comm. in Math. Phys., 199 (1998), 351.   Google Scholar

[19]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 317.  doi: 10.1017/S096249290100006X.  Google Scholar

[20]

S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control,, J. Geom. Phys., 38 (2001), 343.  doi: 10.1016/S0393-0440(00)00069-3.  Google Scholar

[21]

P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417.   Google Scholar

[22]

J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories,, Rep. Math. Phys., 56 (2005), 387.  doi: 10.1016/S0034-4877(05)80093-X.  Google Scholar

[23]

J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids,, J. Geom. Phys., 57 (2007), 665.  doi: 10.1016/j.geomphys.2006.05.006.  Google Scholar

[24]

M. West, "Variational Integrators,'', Ph.D. Thesis, (2004).   Google Scholar

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