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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 |
References:
[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
doi: 10.1063/1.2008214. |
[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.
doi: 10.1016/0393-0440(95)00016-X. |
[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.
doi: 10.1016/j.amc.2005.11.002. |
[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.
doi: 10.1016/j.geomphys.2005.04.002. |
[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
doi: 10.1016/S0034-4877(03)80006-X. |
[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.
doi: 10.1016/0370-2693(83)90687-1. |
[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.
doi: 10.1016/S0393-0440(00)00004-8. |
[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.
doi: 10.1017/S096249290100006X. |
[20] |
S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control, J. Geom. Phys., 38 (2001), 343-365.
doi: 10.1016/S0393-0440(00)00069-3. |
[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.
doi: 10.1016/S0034-4877(05)80093-X. |
[23] |
J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids, J. Geom. Phys., 57 (2007), 665-689.
doi: 10.1016/j.geomphys.2006.05.006. |
[24] |
M. West, "Variational Integrators,'' Ph.D. Thesis, California Institute of Technology, 2004. |
show all references
References:
[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
doi: 10.1063/1.2008214. |
[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.
doi: 10.1016/0393-0440(95)00016-X. |
[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.
doi: 10.1016/j.amc.2005.11.002. |
[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.
doi: 10.1016/j.geomphys.2005.04.002. |
[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
doi: 10.1016/S0034-4877(03)80006-X. |
[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.
doi: 10.1016/0370-2693(83)90687-1. |
[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.
doi: 10.1016/S0393-0440(00)00004-8. |
[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.
doi: 10.1017/S096249290100006X. |
[20] |
S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control, J. Geom. Phys., 38 (2001), 343-365.
doi: 10.1016/S0393-0440(00)00069-3. |
[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.
doi: 10.1016/S0034-4877(05)80093-X. |
[23] |
J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids, J. Geom. Phys., 57 (2007), 665-689.
doi: 10.1016/j.geomphys.2006.05.006. |
[24] |
M. West, "Variational Integrators,'' Ph.D. Thesis, California Institute of Technology, 2004. |
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