Abstract
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the structural properties of these systems, like the symplectic form, the evolution of the momentum maps or the energy behaviour. Also they are easily applicable to optimal control problems based on mechanical systems as proposed in Ober-Blöbaum et al. (ESAIM Control Optim Calc Var 17(2):322–352, 2011).Following the same approach, we develop a family of variational integrators to which we refer as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta. These two families of integrators are, in principle and by construction, different one from the other. Furthermore, the symplectic Galerkin family can as easily be applied to optimal control problems, for which Campos et al. (Higher order variational time discretization of optimal control problems. In: Proceedings of the 20th international symposium on mathematical theory of networks and systems, Melbourne, 2012) is a particular case.
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References
Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings, Reading (1978)
Campos, C.M., Cendra, H., Díaz, V.A., Martín de Diego, D.: Discrete Lagrange-d’Alembert-Poincaré equations for Euler’s disk. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 106(1), 225–234 (2012)
Campos, C.M., Junge, O., Ober-Blöbaum, S.: Higher order variational time discretization of optimal control problems. In: Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne (2012)
Colombo, L., Jiménez, F., Martín de Diego, D.: Discrete second-order Euler-Poincaré equations. Applications to optimal control. Int. J. Geom. Methods Mod. Phys. 9(4), 1250037 (2012)
Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87(2), 247–282 (2000)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010)
Iglesias, D., Marrero, J.C., de Diego, D.M., Martínez, E.: Discrete nonholonomic Lagrangian systems on Lie groupoids. J. Nonlinear Sci. 18(3), 221–276 (2008)
Johnson, E.R., Murphey, T.D.: Dangers of two-point holonomic constraints for variational integrators. In: Proceedings of American Control Conference, St. Louis, Missouri, USA, June 10–12, 2009. IEEE, pp. 4723–4728. Piscataway (2009). http://www.a2c2.org/conferences/acc2009/
Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discret. Cont. Dyn. Syst. 3(1), 61–84 (2010)
Leok, M.: Foundations of computational geometric mechanics. Ph.D. thesis, California Institute of Technology (2004)
Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)
Ober-Blöbaum, S., Junge, O., Marsden, J.: Discrete mechanics and optimal control: an analysis. ESAIM Control Optim. Calc. Var. 17(2), 322–352 (2011)
Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Applied Mathematics and Mathematical Computation, vol. 7. Chapman & Hall, London (1994)
Suris, Y.B.: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2(4), 78–87 (1990)
Veselov. A.P.: Integrable systems with discrete time, and difference operators. Funct. Anal. Appl. 22(2), 83–93 (1988)
Acknowledgements
This work has been partially supported by MEC (Spain) Grants MTM2010-21186-C02-01, MTM2011-15725E, the ICMAT Severo Ochoa project SEV-2011-0087 and the European project IRSES-project “Geomech-246981”. Besides, the author wants to specially thank professors Sina Ober-Blöbaum, from the University of Paderborn, and David Martín de Diego, from the Instituto de Ciencias Matemáticas, for fruitful conversations on the subject and their constant support.
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Campos, C.M. (2014). High Order Variational Integrators: A Polynomial Approach. In: Casas, F., Martínez, V. (eds) Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-06953-1_24
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DOI: https://doi.org/10.1007/978-3-319-06953-1_24
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