High Order Variational Integrators: A Polynomial Approach

  • Cédric M. CamposEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 4)


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.


Optimal Control Problem Variational Integrator Collocation Point Variational Derivation Polynomial Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings, Reading (1978)zbMATHGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87(2), 247–282 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010)Google Scholar
  7. 7.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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).
  9. 9.
    Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discret. Cont. Dyn. Syst. 3(1), 61–84 (2010)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Leok, M.: Foundations of computational geometric mechanics. Ph.D. thesis, California Institute of Technology (2004)Google Scholar
  11. 11.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ober-Blöbaum, S., Junge, O., Marsden, J.: Discrete mechanics and optimal control: an analysis. ESAIM Control Optim. Calc. Var. 17(2), 322–352 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Applied Mathematics and Mathematical Computation, vol. 7. Chapman & Hall, London (1994)Google Scholar
  14. 14.
    Suris, Y.B.: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2(4), 78–87 (1990)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Veselov. A.P.: Integrable systems with discrete time, and difference operators. Funct. Anal. Appl. 22(2), 83–93 (1988)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Instituto de Ciencias MatemáticasCampus de CantoblancoMadridSpain

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