The Journal of Geometric Mechanics (JGM)

Variational integrators for discrete Lagrange problems
Pages: 343 - 374, Volume 2, Issue 4, December 2010

doi:10.3934/jgm.2010.2.343      Abstract        References        Full text (600.0K)           Related Articles

Pedro L. García - Department of Mathematics, University of Salamanca, Salamanca 37008, Spain (email)
Antonio Fernández - Department of Applied Mathematics, University of Salamanca, Salamanca 37008, Spain (email)
César Rodrigo - CINAMIL, Academia Militar, Amadora 2720-113, Portugal (email)

1 V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, "Dynamical Systems III," Encyclopaedia of Mathematical Sciences, 3, Springer Verlag, Berlin, 1988.       
2 R. Benito and D. Martín de Diego, Discrete vakonomic mechanics, J. Math. Phys., 46 (2005), 083521       
3 A. M. Bloch, "Nonholonomic Mechanics and Control,'' Interdisciplinary Applied Mathematics, 24. Springer Science Ed., 2003.       
4 F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325.       
5 J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energy-momentum integrators, preprint, arXiv:hep-th/0109178.
6 J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics, Appl. Math. Comput., 177 (2006), 226-234.       
7 J. Cortés, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lect. Notes in Math. 1793, Springer Verlag, 2002.       
8 P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610.       
9 P. L. García and C. Rodrigo, Cartan forms and second variation for constrained variational problems, Proceedings of the VII International Conference on Geometry, Integrability and Quantization (Varna, Bulgary) 7 Bulgarian Acad. Sci., Sofia, (2006), 140-153.       
10 H. Goldstein, "Classical Mechanics,'' Addison-Wesley Series in Physics, 1980.       
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), 127-148       
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, version1-2004
13 L. Hsu, Calculus of variations via the Griffiths formalism, J. Diff. Geom., 36 (1992), 551-589.       
14 T. D. Lee, Can time be a discrete dynamical variable?, Phys. Lett. B, 122 (1983), 217-–220.
15 M. de León, D. Martín de Diego and A. Santamaría Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys., 45 (2004).       
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), 251-268       
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), 126-144.       
18 J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. in Math. Phys., 199 (1998), 351-398.       
19 J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 317-514.       
20 S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control, J. Geom. Phys., 38 (2001), 343-365.       
21 P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417-1437.       
22 J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories, Rep. Math. Phys., 56 (2005), 387-411.       
23 J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids, J. Geom. Phys., 57 (2007), 665-689.       
24 M. West, "Variational Integrators,'' Ph.D. Thesis, California Institute of Technology, 2004.       

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